Abstract group 4320.dp: $F_5\times S_3^3$

• Label: 4320.dp
• Order: \( 2^{5} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $60$
• Rank: $4$

Group information

Description:	$F_5\times S_3^3$
Order:	\(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$ x 3, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	383	26	640	4	886	252	1520	104	504	4320
Conjugacy classes	1	15	7	16	1	31	7	38	7	12	135
Divisions	1	15	7	8	1	31	7	19	7	12	108
Autjugacy classes	1	7	3	8	1	9	3	12	3	3	50

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	48	32	16	6	1	135
Irr. rational chars.	16	32	32	20	7	1	108

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$60$
Rank:	$4$
Inequivalent generating quadruples:	$3661432320$

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

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{12}=e^{15}=[a,b]=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(4,7,6), (2,3)(4,7,6)(5,9,8,14)(11,13), (1,3,2)(4,7,6)(5,8)(9,14), (11,12,13) \!\cdots\! \rangle$
Direct product:	$S_3$ ${}^3$ $\, \times\, $ $F_5$
Semidirect product:	$(D_5.S_3^2)$ $\,\rtimes\,$ $D_6$	$(D_5.S_3^2)$ $\,\rtimes\,$ $D_6$	$(D_5.S_3^3)$ $\,\rtimes\,$ $C_2$	$(D_5.S_3^3)$ $\,\rtimes\,$ $C_2$	all 71
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_5\times S_3^2)$ . $D_6$	$(D_5\times S_3^3)$ . $C_2$	$D_5$ . $(C_2\times S_3^3)$	$(C_3\times D_5)$ . $D_6^2$	all 15

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 37952 subgroups in 2244 conjugacy classes, 264 normal (24 characteristic).

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

Special subgroups

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

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 4320: $F_5\times S_3^3$

Order 2160: $D_5.S_3^3$ x 3, $C_3\times F_5\times S_3^2$ x 3, $C_3:S_3^2\times F_5$ x 3, $D_5.S_3^3$ x 3, $D_5.S_3^3$, $F_5\times C_3:S_3^2$, $D_5\times S_3^3$

Order 1440: $D_5.D_6^2$ x 3

Order 1080: $S_3\times C_{15}:C_{12}$ x 6, $C_5:S_3^3$ x 3, $D_{15}\times S_3^2$ x 3, $S_3\times C_{15}:D_6$ x 3, $C_3\times D_5\times S_3^2$ x 3, $C_3^2:(S_3\times F_5)$ x 3, $C_3^2:(S_3\times F_5)$ x 3, $C_3^2:(S_3\times F_5)$ x 3, $S_3\times C_3^2:F_5$ x 3, $C_{15}:(S_3\times C_{12})$ x 3, $C_{15}:(S_3\times C_{12})$ x 3, $S_3\times C_3^2\times F_5$ x 3, $C_5:S_3^3$, $C_5\times S_3^3$, $D_5\times C_3:S_3^2$, $C_3^2:S_3\times F_5$

Order 864: $C_4\times S_3^3$

Order 720: $F_5\times S_3^2$ x 27, $D_{10}.S_3^2$ x 6, $D_{30}:C_{12}$ x 6, $D_{10}\times S_3^2$ x 3, $D_{10}.S_3^2$ x 3, $C_6:S_3\times F_5$ x 3

Order 540: $C_{15}:S_3^2$ x 6, $C_{15}:C_6^2$ x 3, $C_3^3:F_5$ x 3, $C_3^3:F_5$ x 3, $C_{15}:S_3^2$ x 3, $C_{15}:S_3^2$ x 3, $C_{15}:S_3^2$ x 3, $C_{15}:S_3^2$ x 3, $C_{15}:S_3^2$ x 3, $C_{15}:S_3^2$ x 3, $C_{15}\times S_3^2$ x 3, $C_3^3:D_{10}$ x 3, $C_3^3:F_5$, $F_5\times C_3^3$, $C_{15}:S_3^2$, $C_3^3:D_{10}$

Order 480: $C_2\times D_6\times F_5$ x 3

Order 432: $C_{12}:S_3^2$ x 3, $C_{12}\times S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_2.S_3^3$ x 3, $C_2\times S_3^3$, $C_{12}:S_3^2$, $C_2.S_3^3$

Order 360: $D_5.S_3^2$ x 30, $D_{15}:C_{12}$ x 30, $D_5\times S_3^2$ x 27, $C_3:S_3\times F_5$ x 16, $D_5.S_3^2$ x 15, $S_3\times D_{30}$ x 6, $C_{30}:D_6$ x 6, $C_{30}:C_{12}$ x 6, $C_{30}:D_6$ x 3, $C_{10}\times S_3^2$ x 3, $C_{30}:D_6$ x 3, $C_2\times C_3^2:F_5$ x 3, $C_{30}:C_{12}$ x 3

Order 288: $C_2.D_6^2$ x 3

Order 270: $C_3^2:D_{15}$ x 3, $C_3^2:C_{30}$ x 3, $C_3^2\times D_{15}$ x 3, $C_3^2:C_{30}$ x 3, $C_3^2:D_{15}$, $C_3^3:C_{10}$, $D_5\times C_3^3$

Order 240: $D_6\times F_5$ x 39, $D_6\times D_{10}$ x 3, $D_{10}.D_6$ x 3, $D_{10}:C_{12}$ x 3

Order 216: $S_3^3$ x 8, $C_6.S_3^2$ x 6, $C_6:S_3^2$ x 3, $C_6\times S_3^2$ x 3, $C_3:S_3\times C_{12}$ x 3, $C_6.C_6^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6:S_3^2$, $C_4\times C_3^2:S_3$

Order 180: $S_3\times D_{15}$ x 30, $C_{15}:D_6$ x 30, $C_{15}:C_{12}$ x 18, $C_{15}:D_6$ x 16, $C_{15}:D_6$ x 15, $C_5\times S_3^2$ x 15, $C_3^2:F_5$ x 10, $C_3^2\times F_5$ x 10, $C_3\times D_{30}$ x 6, $S_3\times C_{30}$ x 6, $C_3:D_{30}$ x 3, $C_{30}:S_3$ x 3, $C_3^2\times D_{10}$ x 3

Order 160: $C_2^3\times F_5$

Order 144: $C_4\times S_3^2$ x 27, $C_{12}\times D_6$ x 6, $D_6.D_6$ x 6, $D_6^2$ x 3, $C_{12}:D_6$ x 3, $C_2^2.S_3^2$ x 3

Order 135: $C_3^2\times C_{15}$

Order 120: $S_3\times F_5$ x 61, $S_3\times D_{10}$ x 39, $C_{30}:C_4$ x 21, $C_6\times F_5$ x 21, $C_2\times D_{30}$ x 3, $C_{10}\times D_6$ x 3, $C_6\times D_{10}$ x 3

Order 108: $C_3:S_3^2$ x 12, $C_3\times S_3^2$ x 12, $C_3:S_3^2$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $C_{12}:C_2^3$ x 3

Order 90: $C_3\times D_{15}$ x 18, $S_3\times C_{15}$ x 18, $C_3:D_{15}$ x 10, $C_{15}:S_3$ x 10, $C_3^2\times D_5$ x 10, $C_3\times C_{30}$ x 3

Order 80: $C_2^2\times F_5$ x 14, $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 39, $S_3\times C_{12}$ x 30, $C_6.D_6$ x 30, $C_{12}:S_3$ x 16, $C_6.D_6$ x 15, $C_6\times D_6$ x 6, $C_6:C_{12}$ x 6, $C_6:D_6$ x 3, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3

Order 60: $S_3\times D_5$ x 61, $D_{30}$ x 21, $S_3\times C_{10}$ x 21, $C_3\times D_{10}$ x 21, $C_{15}:C_4$ x 19, $C_3\times F_5$ x 19, $C_2\times C_{30}$ x 3

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $C_4\times D_6$ x 39, $C_2^2\times D_6$ x 3, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3

Order 45: $C_3\times C_{15}$ x 7

Order 40: $C_2\times F_5$ x 28, $C_2\times D_{10}$ x 14, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 60, $C_6\times S_3$ x 42, $C_6:S_3$ x 22, $C_3:C_{12}$ x 18, $C_3\times C_{12}$ x 10, $C_3^2:C_4$ x 10, $C_6^2$ x 3

Order 32: $C_2^3\times C_4$

Order 30: $D_{15}$ x 19, $C_3\times D_5$ x 19, $C_5\times S_3$ x 19, $C_{30}$ x 12

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 61, $C_2\times D_6$ x 45, $C_2\times C_{12}$ x 21, $C_6:C_4$ x 21, $C_2^2\times C_6$ x 3

Order 20: $D_{10}$ x 28, $F_5$ x 8, $C_2\times C_{10}$ x 7

Order 18: $C_3\times S_3$ x 36, $C_3:S_3$ x 20, $C_3\times C_6$ x 13

Order 16: $C_2^2\times C_4$ x 14, $C_2^4$

Order 15: $C_{15}$ x 7

Order 12: $D_6$ x 103, $C_2\times C_6$ x 24, $C_{12}$ x 19, $C_3:C_4$ x 19

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 15

Order 6: $S_3$ x 38, $C_6$ x 31

Order 5: $C_5$

Order 4: $C_2^2$ x 35, $C_4$ x 8

Order 3: $C_3$ x 7

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4320: $F_5\times S_3^3$

Order 2160: $D_5.S_3^3$, $C_3\times F_5\times S_3^2$, $D_5.S_3^3$, $F_5\times C_3:S_3^2$, $C_3:S_3^2\times F_5$, $D_5\times S_3^3$, $D_5.S_3^3$

Order 1440: $D_5.D_6^2$

Order 1080: $C_5:S_3^3$, $C_5:S_3^3$, $D_{15}\times S_3^2$, $C_5\times S_3^3$, $D_5\times C_3:S_3^2$, $S_3\times C_{15}:D_6$, $C_3\times D_5\times S_3^2$, $C_3^2:(S_3\times F_5)$, $C_3^2:(S_3\times F_5)$, $C_3^2:(S_3\times F_5)$, $S_3\times C_3^2:F_5$, $C_3^2:S_3\times F_5$, $C_{15}:(S_3\times C_{12})$, $S_3\times C_{15}:C_{12}$, $C_{15}:(S_3\times C_{12})$, $S_3\times C_3^2\times F_5$

Order 864: $C_4\times S_3^3$

Order 720: $F_5\times S_3^2$ x 7, $D_{10}\times S_3^2$, $D_{10}.S_3^2$, $D_{10}.S_3^2$, $C_6:S_3\times F_5$, $D_{30}:C_{12}$

Order 540: $C_{15}:C_6^2$, $C_3^3:F_5$, $C_3^3:F_5$, $C_3^3:F_5$, $F_5\times C_3^3$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$, $C_3^3:D_{10}$

Order 480: $C_2\times D_6\times F_5$

Order 432: $C_2\times S_3^3$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}\times S_3^2$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$

Order 360: $D_5\times S_3^2$ x 7, $D_5.S_3^2$ x 6, $C_3:S_3\times F_5$ x 6, $D_{15}:C_{12}$ x 6, $D_5.S_3^2$ x 4, $C_{30}:D_6$, $S_3\times D_{30}$, $C_{10}\times S_3^2$, $C_{30}:D_6$, $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$, $C_{30}:C_{12}$

Order 288: $C_2.D_6^2$

Order 270: $C_3^2:D_{15}$, $C_3^3:C_{10}$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

Order 240: $D_6\times F_5$ x 9, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$

Order 216: $S_3^3$ x 4, $C_6:S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$, $C_4\times C_3^2:S_3$, $C_3:S_3\times C_{12}$, $C_6.C_6^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 180: $S_3\times D_{15}$ x 6, $C_{15}:D_6$ x 6, $C_{15}:D_6$ x 6, $C_{15}:D_6$ x 4, $C_5\times S_3^2$ x 4, $C_3^2:F_5$ x 4, $C_{15}:C_{12}$ x 4, $C_3^2\times F_5$ x 4, $C_3:D_{30}$, $C_{30}:S_3$, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$

Order 160: $C_2^3\times F_5$

Order 144: $C_4\times S_3^2$ x 7, $D_6^2$, $C_{12}:D_6$, $C_{12}\times D_6$, $C_2^2.S_3^2$, $D_6.D_6$

Order 135: $C_3^2\times C_{15}$

Order 120: $S_3\times F_5$ x 15, $S_3\times D_{10}$ x 9, $C_{30}:C_4$ x 5, $C_6\times F_5$ x 5, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$

Order 108: $C_3:S_3^2$ x 4, $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2\times C_{12}$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $C_{12}:C_2^3$

Order 90: $C_3:D_{15}$ x 4, $C_{15}:S_3$ x 4, $C_3\times D_{15}$ x 4, $S_3\times C_{15}$ x 4, $C_3^2\times D_5$ x 4, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$ x 6, $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 10, $C_{12}:S_3$ x 6, $S_3\times C_{12}$ x 6, $C_6.D_6$ x 6, $C_6.D_6$ x 4, $C_6:D_6$, $C_6\times D_6$, $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$

Order 60: $S_3\times D_5$ x 15, $C_{15}:C_4$ x 6, $C_3\times F_5$ x 6, $D_{30}$ x 5, $S_3\times C_{10}$ x 5, $C_3\times D_{10}$ x 5, $C_2\times C_{30}$

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$

Order 48: $C_4\times D_6$ x 9, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 45: $C_3\times C_{15}$ x 3

Order 40: $C_2\times F_5$ x 9, $C_2\times D_{10}$ x 6, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 14, $C_6:S_3$ x 8, $C_6\times S_3$ x 8, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_3:C_{12}$ x 4, $C_6^2$

Order 32: $C_2^3\times C_4$

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 27: $C_3^3$

Order 24: $C_4\times S_3$ x 15, $C_2\times D_6$ x 11, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 5, $C_2^2\times C_6$

Order 20: $D_{10}$ x 9, $F_5$ x 4, $C_2\times C_{10}$ x 3

Order 18: $C_3:S_3$ x 8, $C_3\times S_3$ x 8, $C_3\times C_6$ x 5

Order 16: $C_2^2\times C_4$ x 6, $C_2^4$

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 25, $C_2\times C_6$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 6

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 9, $C_2^3$ x 7

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

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4320: $F_5\times S_3^3$ ($C_1$)

Order 2160: $D_5.S_3^3$ ($C_2$) x 3, $C_3\times F_5\times S_3^2$ ($C_2$) x 3, $C_3:S_3^2\times F_5$ ($C_2$) x 3, $D_5.S_3^3$ ($C_2$) x 3, $D_5.S_3^3$ ($C_2$), $F_5\times C_3:S_3^2$ ($C_2$), $D_5\times S_3^3$ ($C_2$)

Order 1080: $S_3\times C_{15}:C_{12}$ ($C_2^2$) x 6, $C_5:S_3^3$ ($C_4$) x 3, $D_{15}\times S_3^2$ ($C_4$) x 3, $S_3\times C_{15}:D_6$ ($C_2^2$) x 3, $C_3\times D_5\times S_3^2$ ($C_2^2$) x 3, $C_3^2:(S_3\times F_5)$ ($C_2^2$) x 3, $C_3^2:(S_3\times F_5)$ ($C_2^2$) x 3, $C_3^2:(S_3\times F_5)$ ($C_2^2$) x 3, $S_3\times C_3^2:F_5$ ($C_2^2$) x 3, $C_{15}:(S_3\times C_{12})$ ($C_2^2$) x 3, $C_{15}:(S_3\times C_{12})$ ($C_2^2$) x 3, $S_3\times C_3^2\times F_5$ ($C_2^2$) x 3, $C_5:S_3^3$ ($C_4$), $C_5\times S_3^3$ ($C_4$), $D_5\times C_3:S_3^2$ ($C_2^2$), $C_3^2:S_3\times F_5$ ($C_2^2$)

Order 720: $F_5\times S_3^2$ ($S_3$) x 3

Order 540: $C_{15}:S_3^2$ ($C_2\times C_4$) x 6, $C_{15}:C_6^2$ ($C_2^3$) x 3, $C_3^3:F_5$ ($C_2^3$) x 3, $C_3^3:F_5$ ($C_2^3$) x 3, $C_{15}:S_3^2$ ($C_2\times C_4$) x 3, $C_{15}:S_3^2$ ($C_2\times C_4$) x 3, $C_{15}:S_3^2$ ($C_2\times C_4$) x 3, $C_{15}:S_3^2$ ($C_2\times C_4$) x 3, $C_{15}:S_3^2$ ($C_2\times C_4$) x 3, $C_{15}:S_3^2$ ($C_2\times C_4$) x 3, $C_{15}\times S_3^2$ ($C_2\times C_4$) x 3, $C_3^3:D_{10}$ ($C_2^3$) x 3, $C_3^3:F_5$ ($C_2^3$), $F_5\times C_3^3$ ($C_2^3$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_3^3:D_{10}$ ($C_2^3$)

Order 360: $D_5.S_3^2$ ($D_6$) x 6, $D_{15}:C_{12}$ ($D_6$) x 6, $D_5\times S_3^2$ ($D_6$) x 3, $D_5.S_3^2$ ($D_6$) x 3, $C_3:S_3\times F_5$ ($D_6$) x 3

Order 270: $C_3^2:D_{15}$ ($C_2^2\times C_4$) x 3, $C_3^2:C_{30}$ ($C_2^2\times C_4$) x 3, $C_3^2\times D_{15}$ ($C_2^2\times C_4$) x 3, $C_3^2:C_{30}$ ($C_2^2\times C_4$) x 3, $C_3^2:D_{15}$ ($C_2^2\times C_4$), $C_3^3:C_{10}$ ($C_2^2\times C_4$), $D_5\times C_3^3$ ($C_2^4$)

Order 216: $S_3^3$ ($F_5$)

Order 180: $S_3\times D_{15}$ ($C_4\times S_3$) x 6, $C_{15}:D_6$ ($C_2\times D_6$) x 6, $C_{15}:C_{12}$ ($C_2\times D_6$) x 6, $C_{15}:D_6$ ($C_4\times S_3$) x 3, $C_5\times S_3^2$ ($C_4\times S_3$) x 3, $C_{15}:D_6$ ($C_2\times D_6$) x 3, $C_3^2:F_5$ ($C_2\times D_6$) x 3, $C_3^2\times F_5$ ($C_2\times D_6$) x 3

Order 135: $C_3^2\times C_{15}$ ($C_2^3\times C_4$)

Order 120: $S_3\times F_5$ ($S_3^2$) x 3

Order 108: $C_3:S_3^2$ ($C_2\times F_5$) x 3, $C_3\times S_3^2$ ($C_2\times F_5$) x 3, $C_3:S_3^2$ ($C_2\times F_5$)

Order 90: $C_3\times D_{15}$ ($C_4\times D_6$) x 6, $S_3\times C_{15}$ ($C_4\times D_6$) x 6, $C_3:D_{15}$ ($C_4\times D_6$) x 3, $C_{15}:S_3$ ($C_4\times D_6$) x 3, $C_3^2\times D_5$ ($C_2^2\times D_6$) x 3

Order 60: $S_3\times D_5$ ($S_3\times D_6$) x 3, $C_{15}:C_4$ ($S_3\times D_6$) x 3, $C_3\times F_5$ ($S_3\times D_6$) x 3

Order 54: $C_3^2:C_6$ ($C_2^2\times F_5$) x 3, $S_3\times C_3^2$ ($C_2^2\times F_5$) x 3, $C_3^2:S_3$ ($C_2^2\times F_5$)

Order 45: $C_3\times C_{15}$ ($C_{12}:C_2^3$) x 3

Order 36: $S_3^2$ ($S_3\times F_5$) x 3

Order 30: $D_{15}$ ($C_4\times S_3^2$) x 3, $C_3\times D_5$ ($D_6^2$) x 3, $C_5\times S_3$ ($C_4\times S_3^2$) x 3

Order 27: $C_3^3$ ($C_2^3\times F_5$)

Order 20: $F_5$ ($S_3^3$)

Order 18: $C_3\times S_3$ ($D_6\times F_5$) x 6, $C_3:S_3$ ($D_6\times F_5$) x 3

Order 15: $C_{15}$ ($C_2.D_6^2$) x 3

Order 10: $D_5$ ($C_2\times S_3^3$)

Order 9: $C_3^2$ ($C_2\times D_6\times F_5$) x 3

Order 6: $S_3$ ($F_5\times S_3^2$) x 3

Order 5: $C_5$ ($C_4\times S_3^3$)

Order 3: $C_3$ ($D_5.D_6^2$) x 3

Order 1: $C_1$ ($F_5\times S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4320: $F_5\times S_3^3$ ($C_1$)

Order 2160: $D_5.S_3^3$ ($C_2$), $C_3\times F_5\times S_3^2$ ($C_2$), $D_5.S_3^3$ ($C_2$), $F_5\times C_3:S_3^2$ ($C_2$), $C_3:S_3^2\times F_5$ ($C_2$), $D_5\times S_3^3$ ($C_2$), $D_5.S_3^3$ ($C_2$)

Order 1080: $C_5:S_3^3$ ($C_4$), $C_5:S_3^3$ ($C_4$), $D_{15}\times S_3^2$ ($C_4$), $C_5\times S_3^3$ ($C_4$), $D_5\times C_3:S_3^2$ ($C_2^2$), $S_3\times C_{15}:D_6$ ($C_2^2$), $C_3\times D_5\times S_3^2$ ($C_2^2$), $C_3^2:(S_3\times F_5)$ ($C_2^2$), $C_3^2:(S_3\times F_5)$ ($C_2^2$), $C_3^2:(S_3\times F_5)$ ($C_2^2$), $S_3\times C_3^2:F_5$ ($C_2^2$), $C_3^2:S_3\times F_5$ ($C_2^2$), $C_{15}:(S_3\times C_{12})$ ($C_2^2$), $S_3\times C_{15}:C_{12}$ ($C_2^2$), $C_{15}:(S_3\times C_{12})$ ($C_2^2$), $S_3\times C_3^2\times F_5$ ($C_2^2$)

Order 720: $F_5\times S_3^2$ ($S_3$)

Order 540: $C_{15}:C_6^2$ ($C_2^3$), $C_3^3:F_5$ ($C_2^3$), $C_3^3:F_5$ ($C_2^3$), $C_3^3:F_5$ ($C_2^3$), $F_5\times C_3^3$ ($C_2^3$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_3^3:D_{10}$ ($C_2^3$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}:S_3^2$ ($C_2\times C_4$), $C_{15}\times S_3^2$ ($C_2\times C_4$), $C_3^3:D_{10}$ ($C_2^3$)

Order 360: $D_5\times S_3^2$ ($D_6$), $D_5.S_3^2$ ($D_6$), $D_5.S_3^2$ ($D_6$), $C_3:S_3\times F_5$ ($D_6$), $D_{15}:C_{12}$ ($D_6$)

Order 270: $C_3^2:D_{15}$ ($C_2^2\times C_4$), $C_3^3:C_{10}$ ($C_2^2\times C_4$), $C_3^2:D_{15}$ ($C_2^2\times C_4$), $C_3^2:C_{30}$ ($C_2^2\times C_4$), $C_3^2\times D_{15}$ ($C_2^2\times C_4$), $C_3^2:C_{30}$ ($C_2^2\times C_4$), $D_5\times C_3^3$ ($C_2^4$)

Order 216: $S_3^3$ ($F_5$)

Order 180: $C_{15}:D_6$ ($C_4\times S_3$), $S_3\times D_{15}$ ($C_4\times S_3$), $C_5\times S_3^2$ ($C_4\times S_3$), $C_{15}:D_6$ ($C_2\times D_6$), $C_{15}:D_6$ ($C_2\times D_6$), $C_3^2:F_5$ ($C_2\times D_6$), $C_{15}:C_{12}$ ($C_2\times D_6$), $C_3^2\times F_5$ ($C_2\times D_6$)

Order 135: $C_3^2\times C_{15}$ ($C_2^3\times C_4$)

Order 120: $S_3\times F_5$ ($S_3^2$)

Order 108: $C_3:S_3^2$ ($C_2\times F_5$), $C_3:S_3^2$ ($C_2\times F_5$), $C_3\times S_3^2$ ($C_2\times F_5$)

Order 90: $C_3:D_{15}$ ($C_4\times D_6$), $C_{15}:S_3$ ($C_4\times D_6$), $C_3\times D_{15}$ ($C_4\times D_6$), $S_3\times C_{15}$ ($C_4\times D_6$), $C_3^2\times D_5$ ($C_2^2\times D_6$)

Order 60: $S_3\times D_5$ ($S_3\times D_6$), $C_{15}:C_4$ ($S_3\times D_6$), $C_3\times F_5$ ($S_3\times D_6$)

Order 54: $C_3^2:S_3$ ($C_2^2\times F_5$), $C_3^2:C_6$ ($C_2^2\times F_5$), $S_3\times C_3^2$ ($C_2^2\times F_5$)

Order 45: $C_3\times C_{15}$ ($C_{12}:C_2^3$)

Order 36: $S_3^2$ ($S_3\times F_5$)

Order 30: $D_{15}$ ($C_4\times S_3^2$), $C_3\times D_5$ ($D_6^2$), $C_5\times S_3$ ($C_4\times S_3^2$)

Order 27: $C_3^3$ ($C_2^3\times F_5$)

Order 20: $F_5$ ($S_3^3$)

Order 18: $C_3:S_3$ ($D_6\times F_5$), $C_3\times S_3$ ($D_6\times F_5$)

Order 15: $C_{15}$ ($C_2.D_6^2$)

Order 10: $D_5$ ($C_2\times S_3^3$)

Order 9: $C_3^2$ ($C_2\times D_6\times F_5$)

Order 6: $S_3$ ($F_5\times S_3^2$)

Order 5: $C_5$ ($C_4\times S_3^3$)

Order 3: $C_3$ ($D_5.D_6^2$)

Order 1: $C_1$ ($F_5\times S_3^3$)

Series

Derived series

$F_5\times S_3^3$

$\rhd$

$C_3^2\times C_{15}$

$\rhd$

$C_1$

Chief series

$F_5\times S_3^3$

$\rhd$

$D_5\times S_3^3$

$\rhd$

$C_3\times D_5\times S_3^2$

$\rhd$

$C_{15}:C_6^2$

$\rhd$

$D_5\times C_3^3$

$\rhd$

$C_3^2\times C_{15}$

$\rhd$

$C_3\times C_{15}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$F_5\times S_3^3$

$\rhd$

$C_3^2\times C_{15}$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $135 \times 135$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $108 \times 108$ rational character table.