Abstract group 3456.qs: $C_3\times Q_8:A_4^2$

• Label: 3456.qs
• Order: \( 2^{7} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_3\times Q_8:A_4^2$
Order:	\(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.C_3^5.C_2^4$, of order \(248832\)\(\medspace = 2^{10} \cdot 3^{5} \)
Composition factors:	$C_2$ x 7, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	79	890	48	2054	384	3456
Conjugacy classes	1	9	26	4	72	20	132
Divisions	1	9	13	4	36	10	73
Autjugacy classes	1	5	4	2	11	3	26

Dimension	1	2	3	4	6	8	9	12	18	24
Irr. complex chars.	27	0	54	27	0	0	15	9	0	0	132
Irr. rational chars.	1	13	6	1	24	13	5	1	5	4	73

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$62400$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{6}=d^{2}=f^{4}=g^{2}=[c,d]=[c,g]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3,2)(4,6,5)(8,14)(9,13,10,11,15,12), (4,5,7), (4,6)(5,7)(8,14)(9,11) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $A_4$ $\, \times\, $ $(Q_8:A_4)$
Semidirect product:	$(C_3\times Q_8)$ $\,\rtimes\,$ $A_4^2$	$Q_8$ $\,\rtimes\,$ $(C_3\times A_4^2)$	$(C_2^5:A_4)$ $\,\rtimes\,$ $C_3^2$	$(C_2^5:A_4)$ $\,\rtimes\,$ $C_3^2$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $(C_6^2:A_4)$	$C_6$ . $(C_2^2:A_4^2)$	$C_2$ . $(C_2^6:C_3^3)$	$(C_6\times A_4)$ . $(C_2^2:A_4)$	all 6

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

Homology

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

Subgroups

There are 22078 subgroups in 1836 conjugacy classes, 90 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2^2:A_4^2$
Commutator:	$G' \simeq$ $D_4:C_2^4$	$G/G' \simeq$ $C_3^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^6:C_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.C_2^5$	$G/\operatorname{Fit} \simeq$ $C_3^2$
Radical:	$R \simeq$ $C_3\times Q_8:A_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4:C_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

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 3456: $C_3\times Q_8:A_4^2$

Order 1152: $Q_8:A_4^2$ x 9, $C_3\times C_2^5:A_4$ x 2, $C_6\times C_2^4:A_4$, $C_2^5:C_6^2$

Order 864: $C_6\times A_4^2$ x 3, $C_6.A_4^2$ x 2, $C_3^2\times Q_8:A_4$

Order 576: $C_2^4:C_6^2$ x 3, $C_2^4.C_6^2$ x 2, $C_6\times Q_8:A_4$

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

Order 384: $C_2^5:A_4$ x 6, $C_2^5:A_4$ x 3, $A_4\times D_4:C_2^2$ x 3, $C_{12}.C_2^5$

Order 288: $C_2\times A_4^2$ x 27, $C_3\times Q_8:A_4$ x 22, $A_4\times \SL(2,3)$ x 18, $C_2^3:C_6^2$ x 7, $C_2^3.C_6^2$ x 6, $C_2^5:C_3^2$ x 6, $C_3\times Q_8:A_4$ x 4, $C_2^3.C_6^2$ x 3, $C_2^3.C_6^2$ x 2, $Q_8:C_6^2$ x 2, $D_4:C_6^2$

Order 216: $C_6^2:C_6$ x 4, $C_3^2\times \SL(2,3)$ x 2

Order 192: $C_2^5:C_6$ x 9, $C_{12}.C_2^4$ x 6, $A_4\times D_4:C_2$ x 6, $C_{12}:C_2^4$ x 3, $C_2^4:A_4$ x 3, $C_{12}.C_2^4$ x 2

Order 144: $A_4^2$ x 27, $C_2^2:C_6^2$ x 16, $C_2^4:C_3^2$ x 6, $C_4:C_6^2$ x 3, $C_4.C_6^2$ x 2, $C_6\times \SL(2,3)$ x 2, $C_{12}\times A_4$ x 2

Order 128: $D_4:C_2^4$

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

Order 96: $C_6.C_2^4$ x 39, $Q_8:A_4$ x 39, $C_{12}:C_2^3$ x 30, $C_2^3\times A_4$ x 21, $C_2^3:A_4$ x 18, $D_4\times A_4$ x 18, $C_6.C_2^4$ x 16, $Q_8:A_4$ x 12, $C_2^3:C_{12}$ x 9, $Q_8\times A_4$ x 6, $C_2^2\times \SL(2,3)$ x 6, $C_2^4\times C_6$ x 4, $C_2^3\times C_{12}$ x 3, $C_6.C_2^4$ x 2

Order 72: $C_6\times A_4$ x 63, $C_3\times \SL(2,3)$ x 28, $D_4\times C_3^2$ x 6, $C_2\times C_6^2$ x 5, $C_6\times C_{12}$ x 3, $Q_8\times C_3^2$ x 2

Order 64: $D_4:C_2^3$ x 6, $D_4\times C_2^3$ x 3, $D_4:C_2^3$ x 2

Order 54: $C_3^2\times C_6$

Order 48: $C_6\times D_4$ x 99, $D_4:C_6$ x 50, $C_2^2\times A_4$ x 48, $C_2^3\times C_6$ x 21, $C_2^2:A_4$ x 18, $C_2^2\times C_{12}$ x 14, $C_2\times \SL(2,3)$ x 6, $C_4\times A_4$ x 6, $C_6\times Q_8$ x 4

Order 36: $C_3\times A_4$ x 54, $C_6^2$ x 10, $C_3\times C_{12}$ x 2

Order 32: $D_4:C_2^2$ x 36, $C_2^2\times D_4$ x 30, $D_4:C_2^2$ x 16, $C_2^5$ x 4, $C_2^3\times C_4$ x 3, $C_2^2\times Q_8$ x 2

Order 27: $C_3^3$

Order 24: $C_2\times A_4$ x 81, $C_2^2\times C_6$ x 66, $C_3\times D_4$ x 54, $C_2\times C_{12}$ x 31, $\SL(2,3)$ x 30, $C_3\times Q_8$ x 14

Order 18: $C_3\times C_6$ x 18

Order 16: $C_2\times D_4$ x 90, $D_4:C_2$ x 44, $C_2^4$ x 21, $C_2^2\times C_4$ x 14, $C_2\times Q_8$ x 4

Order 12: $C_2\times C_6$ x 69, $A_4$ x 54, $C_{12}$ x 10

Order 9: $C_3^2$ x 13

Order 8: $C_2^3$ x 51, $D_4$ x 36, $C_2\times C_4$ x 22, $Q_8$ x 8

Order 6: $C_6$ x 36

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$ x 13

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3456: $C_3\times Q_8:A_4^2$

Order 1152: $Q_8:A_4^2$, $C_3\times C_2^5:A_4$, $C_6\times C_2^4:A_4$, $C_2^5:C_6^2$

Order 864: $C_6\times A_4^2$, $C_3^2\times Q_8:A_4$, $C_6.A_4^2$

Order 576: $C_6\times Q_8:A_4$, $C_2^4.C_6^2$, $C_2^4:C_6^2$

Order 432: $C_3\times A_4^2$

Order 384: $C_{12}.C_2^5$, $C_2^5:A_4$, $C_2^5:A_4$, $A_4\times D_4:C_2^2$

Order 288: $C_3\times Q_8:A_4$ x 5, $C_2^3:C_6^2$ x 3, $C_3\times Q_8:A_4$, $C_2^3.C_6^2$, $Q_8:C_6^2$, $C_2^3.C_6^2$, $C_2^3.C_6^2$, $A_4\times \SL(2,3)$, $C_2^5:C_3^2$, $C_2\times A_4^2$, $D_4:C_6^2$

Order 216: $C_6^2:C_6$ x 2, $C_3^2\times \SL(2,3)$

Order 192: $C_{12}.C_2^4$ x 3, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $C_2^4:A_4$, $A_4\times D_4:C_2$, $C_2^5:C_6$

Order 144: $C_2^2:C_6^2$ x 6, $C_2^4:C_3^2$, $A_4^2$, $C_4.C_6^2$, $C_4:C_6^2$, $C_6\times \SL(2,3)$, $C_{12}\times A_4$

Order 128: $D_4:C_2^4$

Order 108: $C_3^2\times A_4$ x 2

Order 96: $C_6.C_2^4$ x 9, $C_{12}:C_2^3$ x 6, $C_6.C_2^4$ x 4, $Q_8:A_4$ x 4, $C_2^3\times A_4$ x 3, $C_2^4\times C_6$ x 2, $C_2^3:A_4$, $C_6.C_2^4$, $C_2^3\times C_{12}$, $Q_8:A_4$, $Q_8\times A_4$, $C_2^2\times \SL(2,3)$, $D_4\times A_4$, $C_2^3:C_{12}$

Order 72: $C_6\times A_4$ x 10, $C_3\times \SL(2,3)$ x 4, $C_2\times C_6^2$ x 3, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\times C_{12}$

Order 64: $D_4:C_2^3$ x 3, $D_4:C_2^3$, $D_4\times C_2^3$

Order 54: $C_3^2\times C_6$

Order 48: $C_6\times D_4$ x 13, $C_2^3\times C_6$ x 9, $D_4:C_6$ x 8, $C_2^2\times A_4$ x 6, $C_2^2\times C_{12}$ x 4, $C_6\times Q_8$ x 2, $C_2^2:A_4$, $C_2\times \SL(2,3)$, $C_4\times A_4$

Order 36: $C_3\times A_4$ x 7, $C_6^2$ x 6, $C_3\times C_{12}$

Order 32: $D_4:C_2^2$ x 8, $C_2^2\times D_4$ x 6, $D_4:C_2^2$ x 4, $C_2^5$ x 2, $C_2^2\times Q_8$, $C_2^3\times C_4$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 22, $C_2\times A_4$ x 8, $C_2\times C_{12}$ x 7, $C_3\times D_4$ x 5, $C_3\times Q_8$ x 4, $\SL(2,3)$ x 3

Order 18: $C_3\times C_6$ x 7

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 9, $D_4:C_2$ x 7, $C_2^2\times C_4$ x 4, $C_2\times Q_8$ x 2

Order 12: $C_2\times C_6$ x 21, $A_4$ x 5, $C_{12}$ x 3

Order 9: $C_3^2$ x 4

Order 8: $C_2^3$ x 19, $C_2\times C_4$ x 6, $D_4$ x 4, $Q_8$ x 3

Order 6: $C_6$ x 11

Order 4: $C_2^2$ x 15, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3456: $C_3\times Q_8:A_4^2$ ($C_1$)

Order 1152: $Q_8:A_4^2$ ($C_3$) x 9, $C_3\times C_2^5:A_4$ ($C_3$) x 2, $C_6\times C_2^4:A_4$ ($C_3$), $C_2^5:C_6^2$ ($C_3$)

Order 384: $C_2^5:A_4$ ($C_3^2$) x 6, $C_2^5:A_4$ ($C_3^2$) x 3, $A_4\times D_4:C_2^2$ ($C_3^2$) x 3, $C_{12}.C_2^5$ ($C_3^2$)

Order 288: $C_2^3:C_6^2$ ($A_4$) x 3, $C_2^3.C_6^2$ ($A_4$) x 2, $C_3\times Q_8:A_4$ ($A_4$)

Order 128: $D_4:C_2^4$ ($C_3^3$)

Order 96: $C_2^3\times A_4$ ($C_3\times A_4$) x 9, $Q_8\times A_4$ ($C_3\times A_4$) x 6, $C_2^4\times C_6$ ($C_3\times A_4$) x 3, $Q_8:A_4$ ($C_3\times A_4$) x 3, $C_6.C_2^4$ ($C_3\times A_4$) x 2, $C_6.C_2^4$ ($C_3\times A_4$)

Order 72: $C_6\times A_4$ ($C_2^2:A_4$)

Order 36: $C_3\times A_4$ ($Q_8:A_4$)

Order 32: $C_2^5$ ($C_3^2\times A_4$) x 3, $C_2^2\times Q_8$ ($C_3^2\times A_4$) x 2, $D_4:C_2^2$ ($C_3^2\times A_4$)

Order 24: $C_2^2\times C_6$ ($A_4^2$) x 3, $C_2\times A_4$ ($C_2^4:C_3^2$) x 3, $C_3\times Q_8$ ($A_4^2$) x 2, $C_2^2\times C_6$ ($C_2^4:C_3^2$)

Order 12: $A_4$ ($C_3\times Q_8:A_4$) x 3, $C_2\times C_6$ ($C_3\times Q_8:A_4$)

Order 8: $C_2^3$ ($C_3\times A_4^2$) x 3, $Q_8$ ($C_3\times A_4^2$) x 2, $C_2^3$ ($C_6^2:A_4$)

Order 6: $C_6$ ($C_2^2:A_4^2$)

Order 4: $C_2^2$ ($C_3^2\times Q_8:A_4$)

Order 3: $C_3$ ($Q_8:A_4^2$)

Order 2: $C_2$ ($C_2^6:C_3^3$)

Order 1: $C_1$ ($C_3\times Q_8:A_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3456: $C_3\times Q_8:A_4^2$ ($C_1$)

Order 1152: $Q_8:A_4^2$ ($C_3$), $C_3\times C_2^5:A_4$ ($C_3$), $C_6\times C_2^4:A_4$ ($C_3$), $C_2^5:C_6^2$ ($C_3$)

Order 384: $C_{12}.C_2^5$ ($C_3^2$), $C_2^5:A_4$ ($C_3^2$), $C_2^5:A_4$ ($C_3^2$), $A_4\times D_4:C_2^2$ ($C_3^2$)

Order 288: $C_3\times Q_8:A_4$ ($A_4$), $C_2^3.C_6^2$ ($A_4$), $C_2^3:C_6^2$ ($A_4$)

Order 128: $D_4:C_2^4$ ($C_3^3$)

Order 96: $C_2^4\times C_6$ ($C_3\times A_4$), $C_2^3\times A_4$ ($C_3\times A_4$), $C_6.C_2^4$ ($C_3\times A_4$), $C_6.C_2^4$ ($C_3\times A_4$), $Q_8:A_4$ ($C_3\times A_4$), $Q_8\times A_4$ ($C_3\times A_4$)

Order 72: $C_6\times A_4$ ($C_2^2:A_4$)

Order 36: $C_3\times A_4$ ($Q_8:A_4$)

Order 32: $C_2^5$ ($C_3^2\times A_4$), $D_4:C_2^2$ ($C_3^2\times A_4$), $C_2^2\times Q_8$ ($C_3^2\times A_4$)

Order 24: $C_2^2\times C_6$ ($C_2^4:C_3^2$), $C_2^2\times C_6$ ($A_4^2$), $C_2\times A_4$ ($C_2^4:C_3^2$), $C_3\times Q_8$ ($A_4^2$)

Order 12: $C_2\times C_6$ ($C_3\times Q_8:A_4$), $A_4$ ($C_3\times Q_8:A_4$)

Order 8: $C_2^3$ ($C_6^2:A_4$), $C_2^3$ ($C_3\times A_4^2$), $Q_8$ ($C_3\times A_4^2$)

Order 6: $C_6$ ($C_2^2:A_4^2$)

Order 4: $C_2^2$ ($C_3^2\times Q_8:A_4$)

Order 3: $C_3$ ($Q_8:A_4^2$)

Order 2: $C_2$ ($C_2^6:C_3^3$)

Order 1: $C_1$ ($C_3\times Q_8:A_4^2$)

Series

Derived series

$C_3\times Q_8:A_4^2$

$\rhd$

$D_4:C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_3\times Q_8:A_4^2$

$\rhd$

$Q_8:A_4^2$

$\rhd$

$C_2^5:A_4$

$\rhd$

$D_4:C_2^4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_3\times Q_8:A_4^2$

$\rhd$

$D_4:C_2^4$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 15 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 $132 \times 132$ 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 $73 \times 73$ rational character table.