Abstract group 8640.ck: $A_4\times D_6\times A_5$

• Label: 8640.ck
• Order: \( 2^{6} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $14$
• Trans deg.: $90$
• Rank: $2$

Group information

Description:	$A_4\times D_6\times A_5$
Order:	\(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \)
Exponent:	\(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:	$D_6\times S_4\times S_5$, of order \(34560\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $A_5$
Derived length:	$2$

This group is nonabelian, an A-group, and nonsolvable.

Group statistics

Order	1	2	3	5	6	10	15	30
Elements	1	511	566	24	4106	744	624	2064	8640
Conjugacy classes	1	15	11	2	45	14	10	22	120
Divisions	1	15	7	1	31	7	3	7	72
Autjugacy classes	1	11	7	1	26	5	3	6	60

Dimension	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	30	36
Irr. complex chars.	12	6	28	12	12	14	6	8	6	4	4	0	4	0	2	2	0	120
Irr. rational chars.	4	6	4	6	4	6	6	0	6	10	4	2	4	2	4	2	2	72

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$90$
Rank:	$2$
Inequivalent generating pairs:	$228$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	24
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(9,14,13)(10,12,11), (10,11,12), (2,5,6)(3,4,7), (5,7), (2,4)(3,6), (1,9)(8,14)(10,11,12), (2,4)(5,7), (2,3,7,4,6,5)(10,12)\rangle$
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $A_4$ $\, \times\, $ $A_5$
Semidirect product:	$(C_6\times A_4\times A_5)$ $\,\rtimes\,$ $C_2$	$(C_6\times A_5)$ $\,\rtimes\,$ $(C_2\times A_4)$	$(C_6\times A_4)$ $\,\rtimes\,$ $(C_2\times A_5)$	$(C_2\times D_6)$ $\,\rtimes\,$ $(C_6\times A_5)$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 97784 subgroups in 1965 conjugacy classes, 42 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times A_4\times A_5$
Commutator:	$G' \simeq$ $C_2^2\times \GL(2,4)$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_4\times D_6\times A_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6$	$G/\operatorname{Fit} \simeq$ $C_6\times A_5$
Radical:	$R \simeq$ $A_4\times D_6$	$G/R \simeq$ $A_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times \GL(2,4)$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6$
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 8640: $A_4\times D_6\times A_5$

Order 4320: $S_3\times A_4\times A_5$ x 2, $C_6\times A_4\times A_5$

Order 2880: $\GL(2,4):C_2^4$, $C_2^4:\GL(2,4)$

Order 2160: $A_4\times \GL(2,4)$, $D_6\times \GL(2,4)$

Order 1728: $D_6\times A_4^2$

Order 1440: $\GL(2,4):C_2^3$ x 6, $C_2^3:\GL(2,4)$ x 4, $S_3\times A_4\times D_{10}$, $C_2^3\times \GL(2,4)$

Order 1080: $S_3\times \GL(2,4)$ x 2, $C_6\times \GL(2,4)$

Order 960: $C_2^4\times A_5$

Order 864: $S_3\times A_4^2$ x 2, $C_6\times A_4^2$, $D_6^2:C_6$

Order 720: $D_6\times A_5$ x 10, $S_3\times D_5\times A_4$ x 4, $C_2^2\times \GL(2,4)$ x 4, $A_4\times A_5$ x 2, $A_4\times D_{30}$, $S_3\times C_{10}\times A_4$, $C_3\times A_4\times D_{10}$

Order 576: $C_2^2:A_4\times D_6$ x 2, $C_2^2\times A_4\times D_6$ x 2, $C_2^2\times A_4^2$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^3\times A_5$ x 7, $C_{15}:C_2^5$, $C_2\times A_4\times D_{10}$

Order 432: $A_4\times S_3^2$ x 4, $C_6\times S_3\times A_4$ x 3, $C_6:S_3\times A_4$, $C_3\times A_4^2$

Order 360: $C_6\times A_5$ x 7, $S_3\times A_5$ x 4, $A_4\times D_{15}$ x 2, $C_5\times S_3\times A_4$ x 2, $C_3\times D_5\times A_4$ x 2, $A_4\times C_{30}$, $C_{30}:D_6$

Order 288: $C_2\times A_4\times D_6$ x 13, $C_2\times A_4^2$ x 7, $S_3\times C_2^2:A_4$ x 4, $C_2^5:C_3^2$ x 2, $C_2^3:C_6^2$ x 2, $C_2\times D_6^2$

Order 240: $C_2^2\times A_5$ x 13, $D_6\times D_{10}$ x 12, $A_4\times D_{10}$ x 7, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$

Order 216: $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6\times S_3^2$

Order 192: $C_2^2\wr C_3$ x 2, $A_4\times C_2^4$ x 2, $D_6\times C_2^4$

Order 180: $C_{15}:D_6$ x 4, $\GL(2,4)$ x 3, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$, $A_4\times C_{15}$

Order 160: $C_2^3\times D_{10}$

Order 144: $A_4\times D_6$ x 33, $D_6^2$ x 12, $C_2^2:C_6^2$ x 8, $A_4^2$ x 5, $C_6^2:C_2^2$ x 3, $C_2^4:C_3^2$ x 2, $C_6^2:C_2^2$

Order 120: $S_3\times D_{10}$ x 38, $C_6\times D_{10}$ x 7, $C_2\times A_5$ x 7, $C_2\times D_{30}$ x 6, $C_{10}\times D_6$ x 6, $D_5\times A_4$ x 6, $C_{10}\times A_4$ x 4, $C_2^2\times C_{30}$

Order 108: $C_3\times S_3^2$ x 4, $C_3^2\times D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3^2:D_6$

Order 96: $C_2^3\times A_4$ x 16, $C_2^3\times D_6$ x 15, $C_2^3:A_4$ x 8, $C_2^4\times C_6$

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

Order 80: $C_2^2\times D_{10}$ x 14, $C_2^3\times C_{10}$

Order 72: $S_3\times D_6$ x 38, $S_3\times A_4$ x 26, $C_6\times A_4$ x 24, $C_6\times D_6$ x 19, $C_6:D_6$ x 6, $C_2\times C_6^2$ x 2

Order 64: $C_2^6$

Order 60: $S_3\times D_5$ x 24, $C_3\times D_{10}$ x 17, $D_{30}$ x 10, $S_3\times C_{10}$ x 10, $C_2\times C_{30}$ x 4, $C_5\times A_4$ x 2, $A_5$

Order 54: $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 2, $C_3^2\times C_6$

Order 48: $C_2^2\times D_6$ x 99, $C_2^2\times A_4$ x 37, $C_2^3\times C_6$ x 9, $C_2^2:A_4$ x 4

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

Order 40: $C_2\times D_{10}$ x 50, $C_2^2\times C_{10}$ x 7

Order 36: $C_6\times S_3$ x 39, $S_3^2$ x 24, $C_3\times A_4$ x 16, $C_6:S_3$ x 10, $C_6^2$ x 7

Order 32: $C_2^5$ x 15

Order 30: $C_3\times D_5$ x 10, $C_{30}$ x 7, $D_{15}$ x 4, $C_5\times S_3$ x 4

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 208, $C_2\times A_4$ x 40, $C_2^2\times C_6$ x 39

Order 20: $D_{10}$ x 44, $C_2\times C_{10}$ x 13

Order 18: $C_3\times S_3$ x 22, $C_3\times C_6$ x 13, $C_3:S_3$ x 4

Order 16: $C_2^4$ x 91

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 122, $C_2\times C_6$ x 56, $A_4$ x 14

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

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 175

Order 6: $C_6$ x 31, $S_3$ x 20

Order 5: $C_5$

Order 4: $C_2^2$ x 91

Order 3: $C_3$ x 7

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 8640: $A_4\times D_6\times A_5$

Order 4320: $C_6\times A_4\times A_5$, $S_3\times A_4\times A_5$

Order 2880: $\GL(2,4):C_2^4$, $C_2^4:\GL(2,4)$

Order 2160: $A_4\times \GL(2,4)$, $D_6\times \GL(2,4)$

Order 1728: $D_6\times A_4^2$

Order 1440: $\GL(2,4):C_2^3$ x 4, $C_2^3:\GL(2,4)$ x 3, $S_3\times A_4\times D_{10}$, $C_2^3\times \GL(2,4)$

Order 1080: $C_6\times \GL(2,4)$, $S_3\times \GL(2,4)$

Order 960: $C_2^4\times A_5$

Order 864: $C_6\times A_4^2$, $D_6^2:C_6$, $S_3\times A_4^2$

Order 720: $D_6\times A_5$ x 6, $C_2^2\times \GL(2,4)$ x 4, $S_3\times D_5\times A_4$ x 2, $A_4\times A_5$ x 2, $A_4\times D_{30}$, $S_3\times C_{10}\times A_4$, $C_3\times A_4\times D_{10}$

Order 576: $C_2^2\times A_4\times D_6$ x 2, $C_2^2:A_4\times D_6$, $C_2^2\times A_4^2$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^3\times A_5$ x 5, $C_{15}:C_2^5$, $C_2\times A_4\times D_{10}$

Order 432: $C_6\times S_3\times A_4$ x 3, $A_4\times S_3^2$ x 2, $C_6:S_3\times A_4$, $C_3\times A_4^2$

Order 360: $C_6\times A_5$ x 6, $C_3\times D_5\times A_4$ x 2, $S_3\times A_5$ x 2, $A_4\times C_{30}$, $C_{30}:D_6$, $A_4\times D_{15}$, $C_5\times S_3\times A_4$

Order 288: $C_2\times A_4\times D_6$ x 9, $C_2\times A_4^2$ x 5, $C_2^3:C_6^2$ x 2, $C_2^5:C_3^2$, $C_2\times D_6^2$, $S_3\times C_2^2:A_4$

Order 240: $C_2^2\times A_5$ x 9, $D_6\times D_{10}$ x 8, $A_4\times D_{10}$ x 5, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$

Order 216: $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 2, $C_6\times S_3^2$, $C_6^2:C_6$

Order 192: $A_4\times C_2^4$ x 2, $D_6\times C_2^4$, $C_2^2\wr C_3$

Order 180: $\GL(2,4)$ x 3, $C_{15}:D_6$ x 2, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$, $A_4\times C_{15}$

Order 160: $C_2^3\times D_{10}$

Order 144: $A_4\times D_6$ x 21, $C_2^2:C_6^2$ x 8, $D_6^2$ x 8, $A_4^2$ x 4, $C_6^2:C_2^2$ x 3, $C_6^2:C_2^2$, $C_2^4:C_3^2$

Order 120: $S_3\times D_{10}$ x 19, $C_6\times D_{10}$ x 7, $D_5\times A_4$ x 5, $C_2\times A_5$ x 5, $C_2\times D_{30}$ x 4, $C_{10}\times D_6$ x 4, $C_{10}\times A_4$ x 3, $C_2^2\times C_{30}$

Order 108: $C_3^2\times D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3\times S_3^2$ x 2, $C_3^2:D_6$

Order 96: $C_2^3\times A_4$ x 12, $C_2^3\times D_6$ x 11, $C_2^3:A_4$ x 3, $C_2^4\times C_6$

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

Order 80: $C_2^2\times D_{10}$ x 10, $C_2^3\times C_{10}$

Order 72: $C_6\times A_4$ x 19, $S_3\times D_6$ x 19, $C_6\times D_6$ x 15, $S_3\times A_4$ x 13, $C_6:D_6$ x 4, $C_2\times C_6^2$ x 2

Order 64: $C_2^6$

Order 60: $C_3\times D_{10}$ x 14, $S_3\times D_5$ x 10, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_2\times C_{30}$ x 4, $C_5\times A_4$ x 2, $A_5$

Order 54: $S_3\times C_3^2$ x 3, $C_3^2\times C_6$, $C_3^2:C_6$

Order 48: $C_2^2\times D_6$ x 53, $C_2^2\times A_4$ x 27, $C_2^3\times C_6$ x 9, $C_2^2:A_4$ x 2

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

Order 40: $C_2\times D_{10}$ x 29, $C_2^2\times C_{10}$ x 5

Order 36: $C_6\times S_3$ x 28, $C_3\times A_4$ x 13, $S_3^2$ x 10, $C_6^2$ x 7, $C_6:S_3$ x 6

Order 32: $C_2^5$ x 11

Order 30: $C_3\times D_5$ x 9, $C_{30}$ x 6, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 93, $C_2^2\times C_6$ x 31, $C_2\times A_4$ x 25

Order 20: $D_{10}$ x 25, $C_2\times C_{10}$ x 9

Order 18: $C_3\times S_3$ x 15, $C_3\times C_6$ x 12, $C_3:S_3$ x 2

Order 16: $C_2^4$ x 49

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 61, $C_2\times C_6$ x 44, $A_4$ x 10

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

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 79

Order 6: $C_6$ x 26, $S_3$ x 12

Order 5: $C_5$

Order 4: $C_2^2$ x 49

Order 3: $C_3$ x 7

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 8640: $A_4\times D_6\times A_5$ ($C_1$)

Order 4320: $S_3\times A_4\times A_5$ ($C_2$) x 2, $C_6\times A_4\times A_5$ ($C_2$)

Order 2880: $\GL(2,4):C_2^4$ ($C_3$)

Order 2160: $A_4\times \GL(2,4)$ ($C_2^2$)

Order 1440: $\GL(2,4):C_2^3$ ($C_6$) x 2, $C_2^3\times \GL(2,4)$ ($C_6$), $C_2^3:\GL(2,4)$ ($S_3$)

Order 720: $C_2^2\times \GL(2,4)$ ($C_2\times C_6$), $D_6\times A_5$ ($A_4$), $A_4\times A_5$ ($D_6$)

Order 480: $C_2^3\times A_5$ ($C_3\times S_3$)

Order 360: $S_3\times A_5$ ($C_2\times A_4$) x 2, $C_6\times A_5$ ($C_2\times A_4$)

Order 240: $C_2^2\times A_5$ ($C_6\times S_3$)

Order 180: $\GL(2,4)$ ($C_2^2\times A_4$)

Order 144: $A_4\times D_6$ ($A_5$)

Order 120: $C_2\times A_5$ ($S_3\times A_4$)

Order 72: $S_3\times A_4$ ($C_2\times A_5$) x 2, $C_6\times A_4$ ($C_2\times A_5$)

Order 60: $A_5$ ($A_4\times D_6$)

Order 48: $C_2^2\times D_6$ ($\GL(2,4)$)

Order 36: $C_3\times A_4$ ($C_2^2\times A_5$)

Order 24: $C_2\times D_6$ ($C_6\times A_5$) x 2, $C_2^2\times C_6$ ($C_6\times A_5$), $C_2\times A_4$ ($S_3\times A_5$)

Order 12: $C_2\times C_6$ ($C_2^2\times \GL(2,4)$), $D_6$ ($A_4\times A_5$), $A_4$ ($D_6\times A_5$)

Order 8: $C_2^3$ ($S_3\times \GL(2,4)$)

Order 6: $S_3$ ($C_2^3:\GL(2,4)$) x 2, $C_6$ ($C_2^3:\GL(2,4)$)

Order 4: $C_2^2$ ($D_6\times \GL(2,4)$)

Order 3: $C_3$ ($C_2^4:\GL(2,4)$)

Order 2: $C_2$ ($S_3\times A_4\times A_5$)

Order 1: $C_1$ ($A_4\times D_6\times A_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8640: $A_4\times D_6\times A_5$ ($C_1$)

Order 4320: $C_6\times A_4\times A_5$ ($C_2$), $S_3\times A_4\times A_5$ ($C_2$)

Order 2880: $\GL(2,4):C_2^4$ ($C_3$)

Order 2160: $A_4\times \GL(2,4)$ ($C_2^2$)

Order 1440: $C_2^3\times \GL(2,4)$ ($C_6$), $\GL(2,4):C_2^3$ ($C_6$), $C_2^3:\GL(2,4)$ ($S_3$)

Order 720: $C_2^2\times \GL(2,4)$ ($C_2\times C_6$), $D_6\times A_5$ ($A_4$), $A_4\times A_5$ ($D_6$)

Order 480: $C_2^3\times A_5$ ($C_3\times S_3$)

Order 360: $C_6\times A_5$ ($C_2\times A_4$), $S_3\times A_5$ ($C_2\times A_4$)

Order 240: $C_2^2\times A_5$ ($C_6\times S_3$)

Order 180: $\GL(2,4)$ ($C_2^2\times A_4$)

Order 144: $A_4\times D_6$ ($A_5$)

Order 120: $C_2\times A_5$ ($S_3\times A_4$)

Order 72: $C_6\times A_4$ ($C_2\times A_5$), $S_3\times A_4$ ($C_2\times A_5$)

Order 60: $A_5$ ($A_4\times D_6$)

Order 48: $C_2^2\times D_6$ ($\GL(2,4)$)

Order 36: $C_3\times A_4$ ($C_2^2\times A_5$)

Order 24: $C_2^2\times C_6$ ($C_6\times A_5$), $C_2\times D_6$ ($C_6\times A_5$), $C_2\times A_4$ ($S_3\times A_5$)

Order 12: $C_2\times C_6$ ($C_2^2\times \GL(2,4)$), $D_6$ ($A_4\times A_5$), $A_4$ ($D_6\times A_5$)

Order 8: $C_2^3$ ($S_3\times \GL(2,4)$)

Order 6: $C_6$ ($C_2^3:\GL(2,4)$), $S_3$ ($C_2^3:\GL(2,4)$)

Order 4: $C_2^2$ ($D_6\times \GL(2,4)$)

Order 3: $C_3$ ($C_2^4:\GL(2,4)$)

Order 2: $C_2$ ($S_3\times A_4\times A_5$)

Order 1: $C_1$ ($A_4\times D_6\times A_5$)

Series

Derived series

$A_4\times D_6\times A_5$

$\rhd$

$A_4\times D_6\times A_5$

$\rhd$

$C_2^2\times \GL(2,4)$

$\rhd$

$C_2^2\times \GL(2,4)$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$A_4\times D_6\times A_5$

$\rhd$

$A_4\times D_6\times A_5$

$\rhd$

$A_4\times D_6$

$\rhd$

$A_4\times D_6$

$\rhd$

$S_3\times A_4$

$\rhd$

$S_3\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$A_4\times D_6\times A_5$

$\rhd$

$A_4\times D_6\times A_5$

$\rhd$

$C_2^2\times \GL(2,4)$

$\rhd$

$C_2^2\times \GL(2,4)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $72 \times 72$ rational character table.