Abstract group 2880.hl: $S_3\times F_5\times S_4$

• Label: 2880.hl
• Order: \( 2^{6} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $12$
• Trans deg.: $60$
• Rank: $3$

Group information

Description:	$S_3\times F_5\times S_4$
Order:	\(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$S_3\times F_5\times S_4$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2, $C_5$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	239	26	784	4	382	156	872	104	96	168	48	2880
Conjugacy classes	1	11	3	20	1	9	5	16	3	2	3	1	75
Divisions	1	11	3	12	1	9	5	9	3	2	3	1	60
Autjugacy classes	1	11	3	20	1	9	5	16	3	2	3	1	75

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	16	16	16	8	8	4	4	1	2	75
Irr. rational chars.	8	12	8	10	8	5	6	1	2	60

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$60$
Rank:	$3$
Inequivalent generating triples:	$1935360$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	9	9	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{12}=d^{30}=e^{2}=[a,d]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(9,10,12), (6,7,8), (2,3)(4,5)(6,8,7)(9,10,12), (2,4,3,5)(6,8,7)(9,11,10,12) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $F_5$ $\, \times\, $ $S_4$
Semidirect product:	$(A_4\times F_5)$ $\,\rtimes\,$ $D_6$	$(A_4:F_5)$ $\,\rtimes\,$ $D_6$	$A_4$ $\,\rtimes\,$ $(D_6\times F_5)$	$(S_4\times D_{15})$ $\,\rtimes\,$ $C_4$	all 49
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_5\times S_4)$ . $D_6$	$D_5$ . $(D_6\times S_4)$	$(D_6\times D_{10})$ . $D_6$	$(S_3\times D_5\times S_4)$ . $C_2$	all 13
Aut. group:	$\Aut(C_2\times D_{30})$	$\Aut(D_{10}.D_6)$	$\Aut(C_3\times D_5\times A_4)$	$\Aut(A_4\times D_{15})$	all 19

Elements of the group are displayed as permutations of degree 12.

Homology

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

Subgroups

There are 15912 subgroups in 964 conjugacy classes, 70 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3\times F_5\times S_4$
Commutator:	$G' \simeq$ $A_4\times C_{15}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times F_5\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{30}$	$G/\operatorname{Fit} \simeq$ $C_4\times D_6$
Radical:	$R \simeq$ $S_3\times F_5\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_4\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2880: $S_3\times F_5\times S_4$

Order 1440: $S_3\times D_5\times S_4$, $S_3\times A_4\times F_5$, $S_3\times A_4:F_5$, $C_{15}:(C_4\times S_4)$, $C_{15}:C_4\times S_4$, $F_5\times C_3:S_4$, $C_3\times F_5\times S_4$

Order 960: $C_2\times F_5\times S_4$, $S_3\times D_4\times F_5$

Order 720: $S_3\times D_5\times A_4$, $D_5\times C_3:S_4$, $D_{15}:S_4$, $S_4\times D_{15}$, $S_3\times C_5:S_4$, $C_5\times S_3\times S_4$, $C_3\times D_5\times S_4$, $A_4\times C_{15}:C_4$, $C_3\times A_4\times F_5$, $(A_4\times C_{15}):C_4$, $C_3\times A_4:F_5$, $F_5\times S_3^2$

Order 576: $C_4\times S_3\times S_4$

Order 480: $F_5\times S_4$ x 5, $C_2\times D_6\times F_5$ x 2, $C_{15}:(C_4\times D_4)$ x 2, $D_{10}:C_4\times S_3$ x 2, $C_3:D_4\times F_5$ x 2, $D_{60}:C_4$, $C_{20}:C_4\times S_3$, $F_5\times D_{12}$, $D_{15}:C_4^2$, $D_{10}\times S_4$, $C_2\times A_4\times F_5$, $D_{10}.S_4$, $D_{12}:D_{10}$, $C_{15}:C_4\times D_4$, $D_{20}:C_{12}$

Order 360: $D_5.S_3^2$ x 2, $D_{15}:C_{12}$ x 2, $A_4\times D_{15}$, $C_5\times S_3\times A_4$, $C_3\times D_5\times A_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}\times S_4$, $D_5\times S_3^2$, $D_5.S_3^2$, $C_3:S_3\times F_5$

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

Order 288: $C_4\times S_3\times A_4$, $C_{12}:S_4$, $C_{12}\times S_4$, $D_6.S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $D_6\times S_4$

Order 240: $D_6\times F_5$ x 13, $D_5\times S_4$ x 5, $D_{30}:C_4$ x 4, $A_4\times F_5$ x 3, $A_4:F_5$ x 3, $D_{10}.D_6$ x 2, $C_{15}:C_4^2$ x 2, $D_6\times D_{10}$ x 2, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $A_4\times D_{10}$, $C_{10}:S_4$, $C_{10}\times S_4$, $D_4\times D_{15}$, $C_{20}:D_6$, $C_{12}:D_{10}$, $C_{12}:D_{10}$, $S_3\times D_{20}$, $D_5\times D_{12}$, $C_{12}:D_{10}$, $C_{60}:C_4$, $C_{60}:C_4$, $C_{20}:C_{12}$, $C_{12}\times F_5$

Order 192: $C_2^4.D_6$, $C_4^2:D_6$

Order 180: $S_3\times D_{15}$ x 2, $C_{15}:D_6$ x 2, $C_{15}:C_{12}$ x 2, $A_4\times C_{15}$, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$, $C_3^2:F_5$, $C_3^2\times F_5$

Order 160: $D_4\times F_5$ x 8, $C_2^3\times F_5$ x 2, $C_2^3:F_5$ x 2, $D_4\times D_{10}$, $D_{10}.D_4$, $C_{10}:C_4^2$

Order 144: $S_3\times S_4$ x 4, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $C_{12}\times A_4$, $C_4\times S_3^2$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$

Order 120: $S_3\times F_5$ x 18, $S_3\times D_{10}$ x 12, $C_{30}:C_4$ 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, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2, $C_6\times D_{10}$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_5:D_{12}$ x 2, $C_3:D_{20}$ x 2, $C_{15}:D_4$ x 2, $C_6.D_{10}$, $C_6.D_{10}$, $C_{10}\times A_4$, $D_4\times C_{15}$, $D_{60}$, $C_4\times D_{15}$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_3\times D_{20}$, $D_5\times C_{12}$, $D_{15}:C_4$

Order 96: $C_4\times S_4$ x 5, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_{12}:C_2^3$ x 2, $C_{12}:D_4$ x 2, $D_6.D_4$, $C_4\times D_{12}$, $S_3\times C_4^2$, $C_2^2\times S_4$, $D_4\times D_6$, $C_2^3:C_{12}$, $C_2^2.S_4$, $D_4\times C_{12}$, $C_2^3.D_6$, $D_{12}:C_4$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 80: $C_2^2\times F_5$ x 16, $D_4\times D_5$ x 8, $D_{10}:C_4$ x 8, $C_{20}:C_4$ x 4, $C_4\times F_5$ x 4, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 72: $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_6\times A_4$, $S_3\times D_6$, $C_{12}:S_3$, $C_6.D_6$

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

Order 60: $S_3\times D_5$ x 16, $C_{15}:C_4$ x 7, $C_3\times F_5$ x 7, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_5\times A_4$ x 2, $C_2\times C_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 48: $C_4\times D_6$ x 14, $S_3\times D_4$ x 8, $C_2\times S_4$ x 7, $D_6:C_4$ x 4, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_{12}:C_4$ x 3, $C_2^2\times D_6$ x 2, $C_2^2\times C_{12}$ x 2, $C_6:D_4$ x 2, $C_6.C_2^3$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_6.D_4$ x 2, $C_2^2\times A_4$, $C_6\times D_4$, $C_2\times D_{12}$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 25, $C_2\times D_{10}$ x 15, $C_5:D_4$ x 8, $D_{20}$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 36: $S_3^2$ x 4, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6:S_3$, $C_3\times A_4$

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 3

Order 24: $C_4\times S_3$ x 22, $C_2\times D_6$ x 16, $C_2\times C_{12}$ x 8, $C_3:D_4$ x 8, $C_6:C_4$ x 8, $S_4$ x 6, $D_{12}$ x 4, $C_2\times A_4$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2

Order 20: $D_{10}$ x 21, $F_5$ x 8, $C_2\times C_{10}$ x 7, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$

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 3

Order 12: $D_6$ x 28, $C_{12}$ x 9, $C_3:C_4$ x 9, $C_2\times C_6$ x 8, $A_4$ x 2

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 9: $C_3^2$

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

Order 6: $S_3$ x 12, $C_6$ x 9

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2880: $S_3\times F_5\times S_4$

Order 1440: $S_3\times D_5\times S_4$, $S_3\times A_4\times F_5$, $S_3\times A_4:F_5$, $C_{15}:(C_4\times S_4)$, $C_{15}:C_4\times S_4$, $F_5\times C_3:S_4$, $C_3\times F_5\times S_4$

Order 960: $C_2\times F_5\times S_4$, $S_3\times D_4\times F_5$

Order 720: $S_3\times D_5\times A_4$, $D_5\times C_3:S_4$, $D_{15}:S_4$, $S_4\times D_{15}$, $S_3\times C_5:S_4$, $C_5\times S_3\times S_4$, $C_3\times D_5\times S_4$, $A_4\times C_{15}:C_4$, $C_3\times A_4\times F_5$, $(A_4\times C_{15}):C_4$, $C_3\times A_4:F_5$, $F_5\times S_3^2$

Order 576: $C_4\times S_3\times S_4$

Order 480: $F_5\times S_4$ x 5, $C_2\times D_6\times F_5$ x 2, $C_{15}:(C_4\times D_4)$ x 2, $D_{10}:C_4\times S_3$ x 2, $C_3:D_4\times F_5$ x 2, $D_{60}:C_4$, $C_{20}:C_4\times S_3$, $F_5\times D_{12}$, $D_{15}:C_4^2$, $D_{10}\times S_4$, $C_2\times A_4\times F_5$, $D_{10}.S_4$, $D_{12}:D_{10}$, $C_{15}:C_4\times D_4$, $D_{20}:C_{12}$

Order 360: $D_5.S_3^2$ x 2, $D_{15}:C_{12}$ x 2, $A_4\times D_{15}$, $C_5\times S_3\times A_4$, $C_3\times D_5\times A_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}\times S_4$, $D_5\times S_3^2$, $D_5.S_3^2$, $C_3:S_3\times F_5$

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

Order 288: $C_4\times S_3\times A_4$, $C_{12}:S_4$, $C_{12}\times S_4$, $D_6.S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $D_6\times S_4$

Order 240: $D_6\times F_5$ x 13, $D_5\times S_4$ x 5, $D_{30}:C_4$ x 4, $A_4\times F_5$ x 3, $A_4:F_5$ x 3, $D_{10}.D_6$ x 2, $C_{15}:C_4^2$ x 2, $D_6\times D_{10}$ x 2, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $A_4\times D_{10}$, $C_{10}:S_4$, $C_{10}\times S_4$, $D_4\times D_{15}$, $C_{20}:D_6$, $C_{12}:D_{10}$, $C_{12}:D_{10}$, $S_3\times D_{20}$, $D_5\times D_{12}$, $C_{12}:D_{10}$, $C_{60}:C_4$, $C_{60}:C_4$, $C_{20}:C_{12}$, $C_{12}\times F_5$

Order 192: $C_2^4.D_6$, $C_4^2:D_6$

Order 180: $S_3\times D_{15}$ x 2, $C_{15}:D_6$ x 2, $C_{15}:C_{12}$ x 2, $A_4\times C_{15}$, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$, $C_3^2:F_5$, $C_3^2\times F_5$

Order 160: $D_4\times F_5$ x 8, $C_2^3\times F_5$ x 2, $C_2^3:F_5$ x 2, $D_4\times D_{10}$, $D_{10}.D_4$, $C_{10}:C_4^2$

Order 144: $S_3\times S_4$ x 4, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $C_{12}\times A_4$, $C_4\times S_3^2$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$

Order 120: $S_3\times F_5$ x 18, $S_3\times D_{10}$ x 12, $C_{30}:C_4$ 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, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2, $C_6\times D_{10}$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_5:D_{12}$ x 2, $C_3:D_{20}$ x 2, $C_{15}:D_4$ x 2, $C_6.D_{10}$, $C_6.D_{10}$, $C_{10}\times A_4$, $D_4\times C_{15}$, $D_{60}$, $C_4\times D_{15}$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_3\times D_{20}$, $D_5\times C_{12}$, $D_{15}:C_4$

Order 96: $C_4\times S_4$ x 5, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_{12}:C_2^3$ x 2, $C_{12}:D_4$ x 2, $D_6.D_4$, $C_4\times D_{12}$, $S_3\times C_4^2$, $C_2^2\times S_4$, $D_4\times D_6$, $C_2^3:C_{12}$, $C_2^2.S_4$, $D_4\times C_{12}$, $C_2^3.D_6$, $D_{12}:C_4$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

Order 80: $C_2^2\times F_5$ x 16, $D_4\times D_5$ x 8, $D_{10}:C_4$ x 8, $C_{20}:C_4$ x 4, $C_4\times F_5$ x 4, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_4\times D_{10}$

Order 72: $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_6\times A_4$, $S_3\times D_6$, $C_{12}:S_3$, $C_6.D_6$

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

Order 60: $S_3\times D_5$ x 16, $C_{15}:C_4$ x 7, $C_3\times F_5$ x 7, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_5\times A_4$ x 2, $C_2\times C_{30}$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 48: $C_4\times D_6$ x 14, $S_3\times D_4$ x 8, $C_2\times S_4$ x 7, $D_6:C_4$ x 4, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_{12}:C_4$ x 3, $C_2^2\times D_6$ x 2, $C_2^2\times C_{12}$ x 2, $C_6:D_4$ x 2, $C_6.C_2^3$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_6.D_4$ x 2, $C_2^2\times A_4$, $C_6\times D_4$, $C_2\times D_{12}$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 25, $C_2\times D_{10}$ x 15, $C_5:D_4$ x 8, $D_{20}$ x 4, $C_4\times D_5$ x 4, $C_5\times D_4$ x 4, $C_2^2\times C_{10}$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$

Order 36: $S_3^2$ x 4, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6:S_3$, $C_3\times A_4$

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 3

Order 24: $C_4\times S_3$ x 22, $C_2\times D_6$ x 16, $C_2\times C_{12}$ x 8, $C_3:D_4$ x 8, $C_6:C_4$ x 8, $S_4$ x 6, $D_{12}$ x 4, $C_2\times A_4$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2

Order 20: $D_{10}$ x 21, $F_5$ x 8, $C_2\times C_{10}$ x 7, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$

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 3

Order 12: $D_6$ x 28, $C_{12}$ x 9, $C_3:C_4$ x 9, $C_2\times C_6$ x 8, $A_4$ x 2

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 9: $C_3^2$

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

Order 6: $S_3$ x 12, $C_6$ x 9

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2880: $S_3\times F_5\times S_4$ ($C_1$)

Order 1440: $S_3\times D_5\times S_4$ ($C_2$), $S_3\times A_4\times F_5$ ($C_2$), $S_3\times A_4:F_5$ ($C_2$), $C_{15}:(C_4\times S_4)$ ($C_2$), $C_{15}:C_4\times S_4$ ($C_2$), $F_5\times C_3:S_4$ ($C_2$), $C_3\times F_5\times S_4$ ($C_2$)

Order 720: $S_3\times D_5\times A_4$ ($C_2^2$), $D_5\times C_3:S_4$ ($C_2^2$), $D_{15}:S_4$ ($C_4$), $S_4\times D_{15}$ ($C_4$), $S_3\times C_5:S_4$ ($C_4$), $C_5\times S_3\times S_4$ ($C_4$), $C_3\times D_5\times S_4$ ($C_2^2$), $A_4\times C_{15}:C_4$ ($C_2^2$), $C_3\times A_4\times F_5$ ($C_2^2$), $(A_4\times C_{15}):C_4$ ($C_2^2$), $C_3\times A_4:F_5$ ($C_2^2$)

Order 480: $C_2\times D_6\times F_5$ ($S_3$), $F_5\times S_4$ ($S_3$)

Order 360: $A_4\times D_{15}$ ($C_2\times C_4$), $C_5\times S_3\times A_4$ ($C_2\times C_4$), $C_3\times D_5\times A_4$ ($C_2^3$), $C_{15}:S_4$ ($C_2\times C_4$), $C_{15}:S_4$ ($C_2\times C_4$), $C_{15}:S_4$ ($C_2\times C_4$), $C_{15}\times S_4$ ($C_2\times C_4$)

Order 240: $D_6\times D_{10}$ ($D_6$), $D_{10}.D_6$ ($D_6$), $D_{10}:C_{12}$ ($D_6$), $D_5\times S_4$ ($D_6$), $A_4\times F_5$ ($D_6$), $A_4:F_5$ ($D_6$)

Order 180: $A_4\times C_{15}$ ($C_2^2\times C_4$)

Order 144: $S_3\times S_4$ ($F_5$)

Order 120: $C_2\times D_{30}$ ($C_4\times S_3$), $C_{10}\times D_6$ ($C_4\times S_3$), $C_6\times D_{10}$ ($C_2\times D_6$), $D_5\times A_4$ ($C_2\times D_6$), $C_5:S_4$ ($C_4\times S_3$), $C_5\times S_4$ ($C_4\times S_3$), $S_3\times F_5$ ($S_4$)

Order 80: $C_2^2\times F_5$ ($S_3^2$)

Order 72: $S_3\times A_4$ ($C_2\times F_5$), $C_3:S_4$ ($C_2\times F_5$), $C_3\times S_4$ ($C_2\times F_5$)

Order 60: $C_5\times A_4$ ($C_4\times D_6$), $S_3\times D_5$ ($C_2\times S_4$), $C_{15}:C_4$ ($C_2\times S_4$), $C_3\times F_5$ ($C_2\times S_4$), $C_2\times C_{30}$ ($C_4\times D_6$)

Order 40: $C_2\times D_{10}$ ($S_3\times D_6$)

Order 36: $C_3\times A_4$ ($C_2^2\times F_5$)

Order 30: $D_{15}$ ($C_4\times S_4$), $C_3\times D_5$ ($C_2^2\times S_4$), $C_5\times S_3$ ($C_4\times S_4$)

Order 24: $C_2\times D_6$ ($S_3\times F_5$), $S_4$ ($S_3\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times S_3^2$), $F_5$ ($S_3\times S_4$)

Order 15: $C_{15}$ ($C_2^4.D_6$)

Order 12: $C_2\times C_6$ ($D_6\times F_5$), $A_4$ ($D_6\times F_5$)

Order 10: $D_5$ ($D_6\times S_4$)

Order 6: $S_3$ ($F_5\times S_4$)

Order 5: $C_5$ ($C_4\times S_3\times S_4$)

Order 4: $C_2^2$ ($F_5\times S_3^2$)

Order 3: $C_3$ ($C_2\times F_5\times S_4$)

Order 1: $C_1$ ($S_3\times F_5\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2880: $S_3\times F_5\times S_4$ ($C_1$)

Order 1440: $S_3\times D_5\times S_4$ ($C_2$), $S_3\times A_4\times F_5$ ($C_2$), $S_3\times A_4:F_5$ ($C_2$), $C_{15}:(C_4\times S_4)$ ($C_2$), $C_{15}:C_4\times S_4$ ($C_2$), $F_5\times C_3:S_4$ ($C_2$), $C_3\times F_5\times S_4$ ($C_2$)

Order 720: $S_3\times D_5\times A_4$ ($C_2^2$), $D_5\times C_3:S_4$ ($C_2^2$), $D_{15}:S_4$ ($C_4$), $S_4\times D_{15}$ ($C_4$), $S_3\times C_5:S_4$ ($C_4$), $C_5\times S_3\times S_4$ ($C_4$), $C_3\times D_5\times S_4$ ($C_2^2$), $A_4\times C_{15}:C_4$ ($C_2^2$), $C_3\times A_4\times F_5$ ($C_2^2$), $(A_4\times C_{15}):C_4$ ($C_2^2$), $C_3\times A_4:F_5$ ($C_2^2$)

Order 480: $C_2\times D_6\times F_5$ ($S_3$), $F_5\times S_4$ ($S_3$)

Order 360: $A_4\times D_{15}$ ($C_2\times C_4$), $C_5\times S_3\times A_4$ ($C_2\times C_4$), $C_3\times D_5\times A_4$ ($C_2^3$), $C_{15}:S_4$ ($C_2\times C_4$), $C_{15}:S_4$ ($C_2\times C_4$), $C_{15}:S_4$ ($C_2\times C_4$), $C_{15}\times S_4$ ($C_2\times C_4$)

Order 240: $D_6\times D_{10}$ ($D_6$), $D_{10}.D_6$ ($D_6$), $D_{10}:C_{12}$ ($D_6$), $D_5\times S_4$ ($D_6$), $A_4\times F_5$ ($D_6$), $A_4:F_5$ ($D_6$)

Order 180: $A_4\times C_{15}$ ($C_2^2\times C_4$)

Order 144: $S_3\times S_4$ ($F_5$)

Order 120: $C_2\times D_{30}$ ($C_4\times S_3$), $C_{10}\times D_6$ ($C_4\times S_3$), $C_6\times D_{10}$ ($C_2\times D_6$), $D_5\times A_4$ ($C_2\times D_6$), $C_5:S_4$ ($C_4\times S_3$), $C_5\times S_4$ ($C_4\times S_3$), $S_3\times F_5$ ($S_4$)

Order 80: $C_2^2\times F_5$ ($S_3^2$)

Order 72: $S_3\times A_4$ ($C_2\times F_5$), $C_3:S_4$ ($C_2\times F_5$), $C_3\times S_4$ ($C_2\times F_5$)

Order 60: $C_5\times A_4$ ($C_4\times D_6$), $S_3\times D_5$ ($C_2\times S_4$), $C_{15}:C_4$ ($C_2\times S_4$), $C_3\times F_5$ ($C_2\times S_4$), $C_2\times C_{30}$ ($C_4\times D_6$)

Order 40: $C_2\times D_{10}$ ($S_3\times D_6$)

Order 36: $C_3\times A_4$ ($C_2^2\times F_5$)

Order 30: $D_{15}$ ($C_4\times S_4$), $C_3\times D_5$ ($C_2^2\times S_4$), $C_5\times S_3$ ($C_4\times S_4$)

Order 24: $C_2\times D_6$ ($S_3\times F_5$), $S_4$ ($S_3\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times S_3^2$), $F_5$ ($S_3\times S_4$)

Order 15: $C_{15}$ ($C_2^4.D_6$)

Order 12: $C_2\times C_6$ ($D_6\times F_5$), $A_4$ ($D_6\times F_5$)

Order 10: $D_5$ ($D_6\times S_4$)

Order 6: $S_3$ ($F_5\times S_4$)

Order 5: $C_5$ ($C_4\times S_3\times S_4$)

Order 4: $C_2^2$ ($F_5\times S_3^2$)

Order 3: $C_3$ ($C_2\times F_5\times S_4$)

Order 1: $C_1$ ($S_3\times F_5\times S_4$)

Series

Derived series

$S_3\times F_5\times S_4$

$\rhd$

$A_4\times C_{15}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$S_3\times F_5\times S_4$

$\rhd$

$S_3\times D_5\times S_4$

$\rhd$

$C_3\times D_5\times S_4$

$\rhd$

$C_3\times D_5\times A_4$

$\rhd$

$A_4\times C_{15}$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_5\times A_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_3\times F_5\times S_4$

$\rhd$

$A_4\times C_{15}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 23 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $75 \times 75$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.