Abstract group 24000.bk: $D_5:F_5\times S_5$

• Label: 24000.bk
• Order: \( 2^{6} \cdot 3 \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $50$
• Rank: $3$

Group information

Description:	$D_5:F_5\times S_5$
Order:	\(24000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$F_5\wr C_2\times S_5$, of order \(96000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_5$ x 2, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30
Elements	1	935	20	6680	624	1420	3440	4000	480	4320	2080	24000
Conjugacy classes	1	11	1	20	9	7	19	8	4	10	8	98
Divisions	1	11	1	12	9	7	19	4	4	8	8	84
Autjugacy classes	1	8	1	15	5	5	10	6	2	6	4	63

Dimension	1	2	4	5	6	8	10	12	16	20	24	32	40	48
Irr. complex chars.	16	0	24	16	8	4	0	0	8	8	4	4	4	2	98
Irr. rational chars.	8	4	16	8	4	8	4	2	8	8	4	4	4	2	84

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$50$
Rank:	$3$
Inequivalent generating triples:	$40352256$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	32	32
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(1,2)(6,10)(7,8)(12,13)(14,15), (6,13,8,12)(7,11)(9,15,10,14), (1,5,3)(2,4) \!\cdots\! \rangle$
Direct product:	$S_5$ $\, \times\, $ $(D_5:F_5)$
Semidirect product:	$(D_5\times S_5)$ $\,\rtimes\,$ $F_5$	$D_5$ $\,\rtimes\,$ $(F_5\times S_5)$	$A_5$ $\,\rtimes\,$ $(D_{10}:F_5)$	$(D_5^2.S_5)$ $\,\rtimes\,$ $C_2$	all 29
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_5^2\times S_5)$ . $C_2$	$D_5^2$ . $(C_2\times S_5)$	$(A_5\times D_5^2)$ . $C_2^2$	$(A_5:D_5^2)$ . $C_2^2$	all 7

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 166180 subgroups in 1860 conjugacy classes, 52 normal (28 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_5:F_5\times S_5$
Commutator:	$G' \simeq$ $C_5^2\times A_5$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_5:F_5\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^2$	$G/\operatorname{Fit} \simeq$ $C_2\times C_4\times S_5$
Radical:	$R \simeq$ $D_5:F_5$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_5^2\times A_5$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 24000: $D_5:F_5\times S_5$

Order 12000: $C_5:F_5\times S_5$, $C_5:(F_5\times S_5)$, $D_5^2.S_5$, $C_5:(F_5\times S_5)$, $C_5:F_5\times S_5$, $D_5^2\times S_5$, $A_5\times D_5:F_5$

Order 6000: $A_5:D_5^2$ x 2, $C_5\times D_5\times S_5$ x 2, $(C_5\times A_5):F_5$, $C_5:D_5\times S_5$, $A_5\times C_5:F_5$, $A_5\times D_5^2$, $A_5:D_5^2$, $A_5\times C_5:F_5$, $(C_5\times A_5):F_5$

Order 4800: $C_2\times F_5\times S_5$ x 2, $D_{10}^2.D_6$

Order 4000: $D_5:F_5^2$

Order 3000: $C_5\times D_5\times A_5$ x 2, $C_5^2:S_5$ x 2, $C_5^2\times S_5$, $C_5^2:S_5$, $C_5:D_5\times A_5$

Order 2400: $F_5\times S_5$ x 10, $D_{10}\times S_5$ x 2, $C_2\times F_5\times A_5$ x 2, $D_{10}.S_5$ x 2, $C_5:F_5\times S_4$, $S_3\times D_{10}:F_5$, $D_{10}^2.C_6$, $C_5:(F_5\times S_4)$, $C_5:(F_5\times S_4)$, $D_5^2.S_4$, $C_5:F_5\times S_4$, $S_4\times D_5^2$

Order 2000: $F_5\times D_5^2$, $D_5^2:F_5$, $D_5^2:F_5$, $C_5:F_5^2$, $C_5:F_5^2$, $C_5:F_5^2$, $C_5:F_5^2$

Order 1600: $D_{10}^2.C_2^2$

Order 1500: $C_5^2\times A_5$

Order 1200: $D_5\times S_5$ x 10, $D_5^2.D_6$ x 8, $F_5\times A_5$ x 6, $A_5:F_5$ x 6, $A_4:D_5^2$ x 2, $C_{10}^2:D_6$ x 2, $D_{10}\times A_5$ x 2, $C_{10}\times S_5$ x 2, $C_{10}:S_5$ x 2, $D_{30}:F_5$, $D_{30}:F_5$, $D_5^2.D_6$, $S_3\times C_{10}:F_5$, $S_3\times C_{10}:F_5$, $D_5^2:C_{12}$, $A_4\times D_5^2$, $A_4:D_5^2$, $C_{10}^2:D_6$, $C_{10}^2:C_{12}$, $C_{10}^2:C_{12}$, $(A_4\times C_5^2):C_4$, $(A_4\times C_5^2):C_4$, $D_6\times D_5^2$

Order 1000: $C_5^3:(C_2\times C_4)$ x 3, $C_5^3:(C_2\times C_4)$ x 3, $D_5.D_5^2$ x 2, $D_5^2.C_{10}$ x 2, $D_5^3$, $D_5.D_5^2$, $C_5:D_5\times F_5$, $C_5:(D_5\times F_5)$, $C_5:(D_5\times F_5)$, $D_5^2.D_5$, $D_5\times C_5:F_5$, $D_5\times C_5:F_5$, $D_5^2.C_{10}$

Order 960: $C_2\times F_5\times S_4$ x 2, $C_2\times C_4\times S_5$

Order 800: $D_{10}^2.C_2$ x 2, $C_2\times F_5^2$ x 2, $C_{10}^2:(C_2\times C_4)$ x 2, $D_{10}^2.C_2$ x 2, $C_{10}^2:(C_2\times C_4)$ x 2, $D_4\times D_5^2$, $D_4\times C_5:F_5$, $D_4\times C_5:F_5$, $D_{20}:F_5$, $D_5^2.D_4$, $D_{20}:F_5$, $C_4\times D_5:F_5$

Order 600: $S_3\times D_5^2$ x 8, $D_5\times A_5$ x 6, $C_5:S_5$ x 6, $C_5\times S_5$ x 6, $D_{15}:F_5$ x 4, $D_{15}:F_5$ x 4, $D_5^2.S_3$ x 4, $S_3\times C_5:F_5$ x 4, $S_3\times C_5:F_5$ x 4, $D_5^2.C_6$ x 4, $D_5\times D_{30}$ x 2, $C_{30}:D_{10}$ x 2, $C_{10}^2:C_6$ x 2, $C_5^2:S_4$ x 2, $C_{10}\times A_5$ x 2, $C_{30}:D_{10}$, $C_{30}:D_{10}$, $C_6\times D_5^2$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{10}^2:C_6$, $C_5^2:S_4$, $S_4\times C_5^2$

Order 500: $C_5:D_5^2$ x 3, $C_5\times D_5^2$ x 3, $C_5^3:C_4$ x 3, $C_5^2:C_{20}$ x 2, $C_5:D_5^2$, $C_5^3:C_4$, $C_5^3:C_4$, $C_5^3:C_4$, $C_5^3:C_4$, $C_5^2:C_{20}$, $C_5^2:C_{20}$, $F_5\times C_5^2$

Order 480: $F_5\times S_4$ x 10, $C_4\times S_5$ x 4, $C_2\times D_6\times F_5$ x 2, $D_{10}\times S_4$ x 2, $C_2\times A_4\times F_5$ x 2, $D_{10}.S_4$ x 2, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $C_2^2\times S_5$

Order 400: $D_{10}:F_5$ x 14, $F_5^2$ x 10, $D_{10}\times F_5$ x 4, $D_{10}:D_{10}$ x 4, $D_{10}:F_5$ x 4, $D_{10}^2$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}^2:C_4$ x 2, $C_{20}:D_{10}$ x 2, $D_{10}:D_{10}$ x 2, $D_5\times D_{20}$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}.(C_2\times F_5)$ x 2, $C_5^2:C_4^2$ x 2, $C_{20}:D_{10}$, $C_{20}:D_{10}$, $C_4\times D_5^2$, $C_{20}:F_5$, $C_{20}:F_5$, $C_{20}:F_5$, $C_{20}:F_5$

Order 320: $D_{10}.C_2^4$ x 2

Order 300: $D_5\times D_{15}$ x 8, $C_{15}:D_{10}$ x 8, $C_{15}:D_{10}$ x 4, $C_{15}:D_{10}$ x 4, $C_3\times D_5^2$ x 4, $C_5\times A_5$ x 4, $C_5\times D_{30}$ x 2, $C_{15}\times D_{10}$ x 2, $C_{15}:F_5$ x 2, $C_{15}:F_5$ x 2, $C_{15}:F_5$ x 2, $C_{15}:F_5$ x 2, $C_5:D_{30}$, $D_6\times C_5^2$, $C_{15}:D_{10}$, $A_4\times C_5^2$

Order 250: $C_5^2:C_{10}$ x 3, $D_5\times C_5^2$ x 3, $C_5^3:C_2$

Order 240: $D_6\times F_5$ x 26, $D_5\times S_4$ x 10, $A_4\times F_5$ x 6, $A_4:F_5$ x 6, $C_2\times S_5$ x 6, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $D_6\times D_{10}$ x 2, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $A_4\times D_{10}$ x 2, $C_{10}:S_4$ x 2, $C_{10}\times S_4$ x 2, $C_2^2\times A_5$

Order 200: $D_5:F_5$ x 25, $D_5\times F_5$ x 20, $D_5\times D_{10}$ x 13, $C_{10}:F_5$ x 8, $C_{10}:F_5$ x 8, $C_{10}\times D_{10}$ x 4, $C_{10}:F_5$ x 4, $C_{10}\times F_5$ x 4, $C_{10}\wr C_2$ x 4, $C_5:D_{20}$ x 4, $C_{10}:D_{10}$ x 2, $C_{10}^2:C_2$ x 2, $C_5\times D_{20}$ x 2, $D_5\times C_{20}$ x 2, $D_{10}:D_5$ x 2, $C_{10}.D_{10}$ x 2, $D_4\times C_5^2$, $C_5:D_{20}$, $C_{20}:D_5$, $C_{10}.D_{10}$

Order 192: $C_2^4.D_6$

Order 160: $D_4\times F_5$ x 18, $C_2^3\times F_5$ x 4, $C_2^3:F_5$ x 4, $C_{10}:C_4^2$ x 3, $D_4\times D_{10}$ x 2, $D_{10}.D_4$ x 2

Order 150: $D_5\times C_{15}$ x 4, $C_5\times D_{15}$ x 4, $C_{15}:D_5$ x 2, $C_5:D_{15}$ x 2, $S_3\times C_5^2$ x 2, $C_5\times C_{30}$

Order 125: $C_5^3$

Order 120: $S_3\times F_5$ x 40, $S_3\times D_{10}$ x 26, $C_{30}:C_4$ x 14, $C_6\times F_5$ x 14, $D_5\times A_4$ x 6, $C_5:S_4$ x 6, $C_5\times S_4$ x 6, $S_5$ x 4, $C_2\times A_5$ x 3, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2, $C_6\times D_{10}$ x 2, $C_{10}\times A_4$ x 2

Order 100: $D_5^2$ x 23, $C_5\times D_{10}$ x 14, $C_5:F_5$ x 14, $C_5\times F_5$ x 12, $C_5:F_5$ x 12, $C_5:F_5$ x 12, $C_5:D_{10}$ x 7, $C_5:C_{20}$ x 2, $C_{10}^2$ x 2, $C_5\times C_{20}$, $C_5^2:C_4$

Order 96: $C_4\times S_4$ x 4, $C_2^2\times S_4$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 80: $C_2^2\times F_5$ x 38, $D_{10}:C_4$ x 20, $D_4\times D_5$ x 18, $C_4\times F_5$ x 14, $C_{20}:C_4$ x 10, $C_2^2\times D_{10}$ x 4, $C_{10}:D_4$ x 4, $C_4\times D_{10}$ x 3, $D_4\times C_{10}$ x 2, $C_2\times D_{20}$ x 2

Order 75: $C_5\times C_{15}$

Order 64: $C_4^2:C_2^2$

Order 60: $S_3\times D_5$ x 40, $D_{30}$ x 14, $S_3\times C_{10}$ x 14, $C_3\times D_{10}$ x 14, $C_{15}:C_4$ x 12, $C_3\times F_5$ x 12, $C_5\times A_4$ x 4, $C_2\times C_{30}$ x 2, $A_5$

Order 50: $C_5\times D_5$ x 22, $C_5:D_5$ x 13, $C_5\times C_{10}$ x 4

Order 48: $C_4\times D_6$ x 12, $C_2\times S_4$ x 6, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2\times D_6$, $C_2^2\times A_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times F_5$ x 76, $C_2\times D_{10}$ x 35, $C_5:D_4$ x 20, $C_4\times D_5$ x 14, $D_{20}$ x 10, $C_5\times D_4$ x 10, $C_2^2\times C_{10}$ x 4, $C_2\times C_{20}$ x 3, $C_{10}:C_4$ x 3

Order 32: $C_4\times D_4$ x 8, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 30: $D_{15}$ x 12, $C_3\times D_5$ x 12, $C_5\times S_3$ x 12, $C_{30}$ x 8

Order 25: $C_5^2$ x 9

Order 24: $C_4\times S_3$ x 16, $C_2\times D_6$ x 14, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $S_4$ x 4, $C_2\times A_4$ x 3, $C_2^2\times C_6$

Order 20: $D_{10}$ x 60, $F_5$ x 38, $C_2\times C_{10}$ x 19, $C_{20}$ x 8, $C_5:C_4$ x 8

Order 16: $C_2^2\times C_4$ x 17, $C_2\times D_4$ x 12, $C_2^2:C_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 28, $C_2\times C_6$ x 7, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 10: $D_5$ x 28, $C_{10}$ x 19

Order 8: $C_2\times C_4$ x 31, $C_2^3$ x 17, $D_4$ x 16

Order 6: $S_3$ x 8, $C_6$ x 7

Order 5: $C_5$ x 9

Order 4: $C_2^2$ x 28, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 24000: $D_5:F_5\times S_5$

Order 12000: $C_5:F_5\times S_5$, $C_5:(F_5\times S_5)$, $D_5^2.S_5$, $C_5:(F_5\times S_5)$, $C_5:F_5\times S_5$, $D_5^2\times S_5$, $A_5\times D_5:F_5$

Order 6000: $(C_5\times A_5):F_5$, $C_5:D_5\times S_5$, $A_5\times C_5:F_5$, $A_5\times D_5^2$, $A_5:D_5^2$, $A_5:D_5^2$, $C_5\times D_5\times S_5$, $A_5\times C_5:F_5$, $(C_5\times A_5):F_5$

Order 4800: $C_2\times F_5\times S_5$, $D_{10}^2.D_6$

Order 4000: $D_5:F_5^2$

Order 3000: $C_5\times D_5\times A_5$, $C_5^2\times S_5$, $C_5^2:S_5$, $C_5:D_5\times A_5$, $C_5^2:S_5$

Order 2400: $F_5\times S_5$ x 5, $C_5:F_5\times S_4$, $S_3\times D_{10}:F_5$, $D_{10}\times S_5$, $C_2\times F_5\times A_5$, $D_{10}.S_5$, $D_{10}^2.C_6$, $C_5:(F_5\times S_4)$, $C_5:(F_5\times S_4)$, $D_5^2.S_4$, $C_5:F_5\times S_4$, $S_4\times D_5^2$

Order 2000: $F_5\times D_5^2$, $D_5^2:F_5$, $D_5^2:F_5$, $C_5:F_5^2$, $C_5:F_5^2$, $C_5:F_5^2$, $C_5:F_5^2$

Order 1600: $D_{10}^2.C_2^2$

Order 1500: $C_5^2\times A_5$

Order 1200: $D_5^2.D_6$ x 8, $D_5\times S_5$ x 5, $F_5\times A_5$ x 3, $A_5:F_5$ x 3, $D_{30}:F_5$, $D_{30}:F_5$, $D_5^2.D_6$, $S_3\times C_{10}:F_5$, $S_3\times C_{10}:F_5$, $D_5^2:C_{12}$, $A_4\times D_5^2$, $A_4:D_5^2$, $A_4:D_5^2$, $C_{10}^2:D_6$, $C_{10}^2:D_6$, $C_{10}^2:C_{12}$, $C_{10}^2:C_{12}$, $(A_4\times C_5^2):C_4$, $(A_4\times C_5^2):C_4$, $D_{10}\times A_5$, $C_{10}\times S_5$, $C_{10}:S_5$, $D_6\times D_5^2$

Order 1000: $C_5^3:(C_2\times C_4)$ x 2, $C_5^3:(C_2\times C_4)$ x 2, $D_5^3$, $D_5.D_5^2$, $D_5.D_5^2$, $C_5:D_5\times F_5$, $C_5:(D_5\times F_5)$, $C_5:(D_5\times F_5)$, $D_5^2.D_5$, $D_5\times C_5:F_5$, $D_5\times C_5:F_5$, $D_5^2.C_{10}$, $D_5^2.C_{10}$

Order 960: $C_2\times F_5\times S_4$, $C_2\times C_4\times S_5$

Order 800: $D_{10}^2.C_2$ x 2, $C_{10}^2:(C_2\times C_4)$ x 2, $D_{10}^2.C_2$ x 2, $C_{10}^2:(C_2\times C_4)$ x 2, $C_2\times F_5^2$, $D_4\times D_5^2$, $D_4\times C_5:F_5$, $D_4\times C_5:F_5$, $D_{20}:F_5$, $D_5^2.D_4$, $D_{20}:F_5$, $C_4\times D_5:F_5$

Order 600: $S_3\times D_5^2$ x 6, $D_{15}:F_5$ x 4, $D_{15}:F_5$ x 4, $D_5^2.S_3$ x 4, $S_3\times C_5:F_5$ x 4, $S_3\times C_5:F_5$ x 4, $D_5^2.C_6$ x 4, $D_5\times A_5$ x 3, $C_5:S_5$ x 3, $C_5\times S_5$ x 3, $C_{30}:D_{10}$, $D_5\times D_{30}$, $C_{30}:D_{10}$, $C_{30}:D_{10}$, $C_6\times D_5^2$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{30}:F_5$, $C_{10}^2:C_6$, $C_{10}^2:C_6$, $C_5^2:S_4$, $C_5^2:S_4$, $S_4\times C_5^2$, $C_{10}\times A_5$

Order 500: $C_5:D_5^2$ x 2, $C_5\times D_5^2$ x 2, $C_5^3:C_4$ x 2, $C_5:D_5^2$, $C_5^3:C_4$, $C_5^3:C_4$, $C_5^3:C_4$, $C_5^3:C_4$, $C_5^2:C_{20}$, $C_5^2:C_{20}$, $C_5^2:C_{20}$, $F_5\times C_5^2$

Order 480: $F_5\times S_4$ x 5, $C_4\times S_5$ x 4, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $C_2\times D_6\times F_5$, $D_{10}\times S_4$, $C_2\times A_4\times F_5$, $D_{10}.S_4$, $C_2^2\times S_5$

Order 400: $D_{10}:F_5$ x 13, $F_5^2$ x 5, $D_{10}:F_5$ x 4, $D_{10}^2$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}^2:C_4$ x 2, $D_{10}\times F_5$ x 2, $D_{10}:D_{10}$ x 2, $D_{10}:D_{10}$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}^2:C_4$ x 2, $C_{10}.(C_2\times F_5)$ x 2, $C_5^2:C_4^2$ x 2, $C_{20}:D_{10}$, $C_{20}:D_{10}$, $C_{20}:D_{10}$, $D_5\times D_{20}$, $C_4\times D_5^2$, $C_{20}:F_5$, $C_{20}:F_5$, $C_{20}:F_5$, $C_{20}:F_5$

Order 320: $D_{10}.C_2^4$

Order 300: $D_5\times D_{15}$ x 4, $C_{15}:D_{10}$ x 4, $C_{15}:D_{10}$ x 4, $C_{15}:D_{10}$ x 3, $C_3\times D_5^2$ x 3, $C_{15}:F_5$ x 2, $C_{15}:F_5$ x 2, $C_{15}:F_5$ x 2, $C_{15}:F_5$ x 2, $C_5\times A_5$ x 2, $C_5:D_{30}$, $C_5\times D_{30}$, $D_6\times C_5^2$, $C_{15}:D_{10}$, $C_{15}\times D_{10}$, $A_4\times C_5^2$

Order 250: $C_5^2:C_{10}$ x 2, $D_5\times C_5^2$ x 2, $C_5^3:C_2$

Order 240: $D_6\times F_5$ x 13, $D_5\times S_4$ x 5, $C_2\times S_5$ x 4, $A_4\times F_5$ x 3, $A_4:F_5$ x 3, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $A_4\times D_{10}$, $C_{10}:S_4$, $C_{10}\times S_4$, $C_2^2\times A_5$

Order 200: $D_5:F_5$ x 18, $D_5\times F_5$ x 10, $D_5\times D_{10}$ x 9, $C_{10}:F_5$ x 7, $C_{10}:F_5$ x 7, $C_{10}:D_{10}$ x 2, $C_{10}\times D_{10}$ x 2, $C_{10}:F_5$ x 2, $C_{10}\times F_5$ x 2, $C_{10}^2:C_2$ x 2, $C_{10}\wr C_2$ x 2, $C_5:D_{20}$ x 2, $D_4\times C_5^2$, $C_5:D_{20}$, $C_{20}:D_5$, $C_5\times D_{20}$, $D_5\times C_{20}$, $D_{10}:D_5$, $C_{10}.D_{10}$, $C_{10}.D_{10}$

Order 192: $C_2^4.D_6$

Order 160: $D_4\times F_5$ x 9, $C_2^3\times F_5$ x 2, $C_2^3:F_5$ x 2, $C_{10}:C_4^2$ x 2, $D_4\times D_{10}$, $D_{10}.D_4$

Order 150: $C_{15}:D_5$ x 2, $D_5\times C_{15}$ x 2, $C_5:D_{15}$ x 2, $C_5\times D_{15}$ x 2, $S_3\times C_5^2$ x 2, $C_5\times C_{30}$

Order 125: $C_5^3$

Order 120: $S_3\times F_5$ x 20, $S_3\times D_{10}$ x 13, $C_{30}:C_4$ x 7, $C_6\times F_5$ x 7, $D_5\times A_4$ x 3, $C_5:S_4$ x 3, $C_5\times S_4$ x 3, $S_5$ x 3, $C_2\times A_5$ x 2, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_{10}\times A_4$

Order 100: $D_5^2$ x 14, $C_5:F_5$ x 9, $C_5:F_5$ x 8, $C_5\times D_{10}$ x 7, $C_5\times F_5$ x 6, $C_5:D_{10}$ x 6, $C_5:F_5$ x 6, $C_{10}^2$ x 2, $C_5\times C_{20}$, $C_5^2:C_4$, $C_5:C_{20}$

Order 96: $C_4\times S_4$ x 4, $C_2^2\times S_4$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 80: $C_2^2\times F_5$ x 20, $D_{10}:C_4$ x 10, $D_4\times D_5$ x 9, $C_4\times F_5$ x 9, $C_{20}:C_4$ x 5, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $C_4\times D_{10}$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$

Order 75: $C_5\times C_{15}$

Order 64: $C_4^2:C_2^2$

Order 60: $S_3\times D_5$ x 20, $D_{30}$ x 7, $S_3\times C_{10}$ x 7, $C_3\times D_{10}$ x 7, $C_{15}:C_4$ x 6, $C_3\times F_5$ x 6, $C_5\times A_4$ x 2, $A_5$, $C_2\times C_{30}$

Order 50: $C_5\times D_5$ x 11, $C_5:D_5$ x 8, $C_5\times C_{10}$ x 3

Order 48: $C_4\times D_6$ x 12, $C_2\times S_4$ x 4, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2\times D_6$, $C_2^2\times A_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times F_5$ x 40, $C_2\times D_{10}$ x 18, $C_5:D_4$ x 10, $C_4\times D_5$ x 9, $D_{20}$ x 5, $C_5\times D_4$ x 5, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_2^2\times C_{10}$ x 2

Order 32: $C_4\times D_4$ x 8, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 30: $D_{15}$ x 6, $C_3\times D_5$ x 6, $C_5\times S_3$ x 6, $C_{30}$ x 4

Order 25: $C_5^2$ x 5

Order 24: $C_4\times S_3$ x 16, $C_2\times D_6$ x 10, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $S_4$ x 3, $C_2\times A_4$ x 2, $C_2^2\times C_6$

Order 20: $D_{10}$ x 31, $F_5$ x 21, $C_2\times C_{10}$ x 10, $C_{20}$ x 5, $C_5:C_4$ x 5

Order 16: $C_2^2\times C_4$ x 17, $C_2^2:C_4$ x 8, $C_2\times D_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 15: $C_{15}$ x 2

Order 12: $D_6$ x 18, $C_2\times C_6$ x 5, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 10: $D_5$ x 15, $C_{10}$ x 10

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 12, $D_4$ x 10

Order 6: $S_3$ x 6, $C_6$ x 5

Order 5: $C_5$ x 5

Order 4: $C_2^2$ x 20, $C_4$ x 11

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24000: $D_5:F_5\times S_5$ ($C_1$)

Order 12000: $C_5:F_5\times S_5$ ($C_2$), $C_5:(F_5\times S_5)$ ($C_2$), $D_5^2.S_5$ ($C_2$), $C_5:(F_5\times S_5)$ ($C_2$), $C_5:F_5\times S_5$ ($C_2$), $D_5^2\times S_5$ ($C_2$), $A_5\times D_5:F_5$ ($C_2$)

Order 6000: $A_5:D_5^2$ ($C_4$) x 2, $C_5\times D_5\times S_5$ ($C_4$) x 2, $(C_5\times A_5):F_5$ ($C_2^2$), $C_5:D_5\times S_5$ ($C_2^2$), $A_5\times C_5:F_5$ ($C_2^2$), $A_5\times D_5^2$ ($C_2^2$), $A_5:D_5^2$ ($C_2^2$), $A_5\times C_5:F_5$ ($C_2^2$), $(C_5\times A_5):F_5$ ($C_2^2$)

Order 3000: $C_5\times D_5\times A_5$ ($C_2\times C_4$) x 2, $C_5^2:S_5$ ($C_2\times C_4$) x 2, $C_5^2\times S_5$ ($C_2\times C_4$), $C_5^2:S_5$ ($C_2\times C_4$), $C_5:D_5\times A_5$ ($C_2^3$)

Order 1500: $C_5^2\times A_5$ ($C_2^2\times C_4$)

Order 1200: $D_5\times S_5$ ($F_5$) x 2

Order 600: $D_5\times A_5$ ($C_2\times F_5$) x 2, $C_5:S_5$ ($C_2\times F_5$) x 2, $C_5\times S_5$ ($C_2\times F_5$) x 2

Order 300: $C_5\times A_5$ ($C_2^2\times F_5$) x 2

Order 200: $D_5:F_5$ ($S_5$)

Order 120: $S_5$ ($D_5:F_5$)

Order 100: $D_5^2$ ($C_2\times S_5$), $C_5:F_5$ ($C_2\times S_5$), $C_5:F_5$ ($C_2\times S_5$)

Order 60: $A_5$ ($D_{10}:F_5$)

Order 50: $C_5\times D_5$ ($C_4\times S_5$) x 2, $C_5:D_5$ ($C_2^2\times S_5$)

Order 25: $C_5^2$ ($C_2\times C_4\times S_5$)

Order 10: $D_5$ ($F_5\times S_5$) x 2

Order 5: $C_5$ ($C_2\times F_5\times S_5$) x 2

Order 1: $C_1$ ($D_5:F_5\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24000: $D_5:F_5\times S_5$ ($C_1$)

Order 12000: $C_5:F_5\times S_5$ ($C_2$), $C_5:(F_5\times S_5)$ ($C_2$), $D_5^2.S_5$ ($C_2$), $C_5:(F_5\times S_5)$ ($C_2$), $C_5:F_5\times S_5$ ($C_2$), $D_5^2\times S_5$ ($C_2$), $A_5\times D_5:F_5$ ($C_2$)

Order 6000: $(C_5\times A_5):F_5$ ($C_2^2$), $C_5:D_5\times S_5$ ($C_2^2$), $A_5\times C_5:F_5$ ($C_2^2$), $A_5\times D_5^2$ ($C_2^2$), $A_5:D_5^2$ ($C_2^2$), $A_5:D_5^2$ ($C_4$), $C_5\times D_5\times S_5$ ($C_4$), $A_5\times C_5:F_5$ ($C_2^2$), $(C_5\times A_5):F_5$ ($C_2^2$)

Order 3000: $C_5\times D_5\times A_5$ ($C_2\times C_4$), $C_5^2\times S_5$ ($C_2\times C_4$), $C_5^2:S_5$ ($C_2\times C_4$), $C_5:D_5\times A_5$ ($C_2^3$), $C_5^2:S_5$ ($C_2\times C_4$)

Order 1500: $C_5^2\times A_5$ ($C_2^2\times C_4$)

Order 1200: $D_5\times S_5$ ($F_5$)

Order 600: $D_5\times A_5$ ($C_2\times F_5$), $C_5:S_5$ ($C_2\times F_5$), $C_5\times S_5$ ($C_2\times F_5$)

Order 300: $C_5\times A_5$ ($C_2^2\times F_5$)

Order 200: $D_5:F_5$ ($S_5$)

Order 120: $S_5$ ($D_5:F_5$)

Order 100: $D_5^2$ ($C_2\times S_5$), $C_5:F_5$ ($C_2\times S_5$), $C_5:F_5$ ($C_2\times S_5$)

Order 60: $A_5$ ($D_{10}:F_5$)

Order 50: $C_5:D_5$ ($C_2^2\times S_5$), $C_5\times D_5$ ($C_4\times S_5$)

Order 25: $C_5^2$ ($C_2\times C_4\times S_5$)

Order 10: $D_5$ ($F_5\times S_5$)

Order 5: $C_5$ ($C_2\times F_5\times S_5$)

Order 1: $C_1$ ($D_5:F_5\times S_5$)

Series

Derived series

$D_5:F_5\times S_5$

$\rhd$

$D_5:F_5\times S_5$

$\rhd$

$C_5^2\times A_5$

$\rhd$

$C_5^2\times A_5$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$D_5:F_5\times S_5$

$\rhd$

$D_5:F_5\times S_5$

$\rhd$

$A_5\times D_5:F_5$

$\rhd$

$A_5\times D_5:F_5$

$\rhd$

$D_5:F_5$

$\rhd$

$D_5:F_5$

$\rhd$

$C_5:F_5$

$\rhd$

$C_5:F_5$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$D_5:F_5\times S_5$

$\rhd$

$D_5:F_5\times S_5$

$\rhd$

$C_5^2\times A_5$

$\rhd$

$C_5^2\times A_5$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $98 \times 98$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $84 \times 84$ rational character table.