Abstract group 576.8355: $\GL(2,\mathbb{Z}/4):S_3$

• Label: 576.8355
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $11$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$\GL(2,\mathbb{Z}/4):S_3$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^3:D_6^2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian, monomial (hence solvable), and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	123	26	132	174	120	576
Conjugacy classes	1	11	3	8	12	4	39
Divisions	1	11	3	8	12	4	39
Autjugacy classes	1	10	3	7	11	4	36

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	10	8	6	6	1	39
Irr. rational chars.	8	10	8	6	6	1	39

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$40320$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	7	7	7

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{12}=d^{2}=e^{6}=[a,e]=[b,d]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $11$ $\langle(2,4)(6,7), (2,4)(9,10), (1,2)(3,4), (1,3)(2,4), (9,10,11), (5,6,7), (8,9)(10,11), (8,10)(9,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 30 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 19 & 51 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right), \left(\begin{array}{rr} 53 & 12 \\ 36 & 17 \end{array}\right), \left(\begin{array}{rr} 46 & 45 \\ 45 & 26 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Transitive group:	36T761	36T806			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $S_3^2$	$A_4$ $\,\rtimes\,$ $(S_3\times D_4)$	$(D_6\times S_4)$ $\,\rtimes\,$ $C_2$	$(D_6:S_4)$ $\,\rtimes\,$ $C_2$	all 33
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(D_6\times S_4)$	$C_2^3$ . $(S_3\times D_6)$	$(C_6:S_4)$ . $C_2^2$	$C_6$ . $(C_2^2\times S_4)$	all 7

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{60}\Z)$.

Homology

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

Subgroups

There are 4016 subgroups in 500 conjugacy classes, 43 normal (41 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6\times S_4$
Commutator:	$G' \simeq$ $C_6\times A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_6$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $\GL(2,\mathbb{Z}/4):S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 576: $\GL(2,\mathbb{Z}/4):S_3$

Order 288: $A_4\times C_3:D_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2\times C_6:S_4$, $D_6\times S_4$

Order 192: $D_4\times S_4$, $C_2^4:D_6$

Order 144: $C_6:S_4$ x 4, $S_3\times S_4$ x 2, $C_2^2:C_6^2$, $A_4\times D_6$, $C_6\times S_4$, $D_6:D_6$, $A_4\times C_3:C_4$, $A_4:C_{12}$

Order 96: $C_2^2\times S_4$ x 3, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3:D_6$ x 2, $D_4\times D_6$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.D_6$ x 2, $D_4\times A_4$, $C_4:S_4$, $C_4\times S_4$, $C_2^4:C_6$, $C_{12}:D_4$, $D_6:D_4$

Order 72: $C_3:S_4$ x 3, $C_6\times A_4$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6:D_6$, $S_3\times D_6$, $S_3\times A_4$, $C_3\times S_4$, $C_6.D_6$

Order 64: $D_4^2$

Order 48: $C_6:D_4$ x 14, $C_2\times S_4$ x 13, $S_3\times D_4$ x 9, $D_6:C_4$ x 4, $C_2^2\times D_6$ x 3, $C_2^2\times A_4$ x 3, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_2^2:C_{12}$ x 3, $C_6.C_2^3$ x 2, $C_4\times D_6$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_4\times A_4$, $A_4:C_4$, $C_6.D_4$, $C_{12}:C_4$

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

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 24: $C_3:D_4$ x 21, $C_2\times D_6$ x 19, $S_4$ x 8, $C_6:C_4$ x 6, $D_{12}$ x 6, $C_2^2\times C_6$ x 6, $C_2\times A_4$ x 6, $C_3\times D_4$ x 6, $C_4\times S_3$ x 5, $C_2\times C_{12}$ x 3

Order 18: $C_3:S_3$ x 3, $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$ x 28, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 34, $C_2\times C_6$ x 15, $C_3:C_4$ x 6, $C_{12}$ x 4, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 29, $C_2^3$ x 22, $C_2\times C_4$ x 13

Order 6: $S_3$ x 13, $C_6$ x 12

Order 4: $C_2^2$ x 33, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $\GL(2,\mathbb{Z}/4):S_3$

Order 288: $A_4\times C_3:D_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2\times C_6:S_4$, $D_6\times S_4$

Order 192: $D_4\times S_4$, $C_2^4:D_6$

Order 144: $C_6:S_4$ x 3, $C_2^2:C_6^2$, $A_4\times D_6$, $C_6\times S_4$, $S_3\times S_4$, $D_6:D_6$, $A_4\times C_3:C_4$, $A_4:C_{12}$

Order 96: $C_2^2\times S_4$ x 3, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3:D_6$ x 2, $D_4\times D_6$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.D_6$ x 2, $D_4\times A_4$, $C_4:S_4$, $C_4\times S_4$, $C_2^4:C_6$, $C_{12}:D_4$, $D_6:D_4$

Order 72: $C_6\times A_4$ x 2, $C_3:S_4$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6:D_6$, $S_3\times D_6$, $S_3\times A_4$, $C_3\times S_4$, $C_6.D_6$

Order 64: $D_4^2$

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

Order 36: $C_6:S_3$ x 3, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$, $C_3\times A_4$, $S_3^2$

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 24: $C_2\times D_6$ x 17, $C_3:D_4$ x 16, $C_2^2\times C_6$ x 6, $C_3\times D_4$ x 6, $C_6:C_4$ x 5, $D_{12}$ x 5, $C_2\times A_4$ x 5, $S_4$ x 5, $C_4\times S_3$ x 4, $C_2\times C_{12}$ x 3

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

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 24, $C_2\times C_6$ x 15, $C_3:C_4$ x 5, $C_{12}$ x 4, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 23, $C_2^3$ x 20, $C_2\times C_4$ x 11

Order 6: $C_6$ x 11, $S_3$ x 9

Order 4: $C_2^2$ x 28, $C_4$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $\GL(2,\mathbb{Z}/4):S_3$ ($C_1$)

Order 288: $A_4\times C_3:D_4$ ($C_2$), $C_3\times \GL(2,\mathbb{Z}/4)$ ($C_2$), $D_6:S_4$ ($C_2$), $C_2^3.S_3^2$ ($C_2$), $C_2^3.S_3^2$ ($C_2$), $C_2\times C_6:S_4$ ($C_2$), $D_6\times S_4$ ($C_2$)

Order 144: $C_6:S_4$ ($C_2^2$) x 2, $C_2^2:C_6^2$ ($C_2^2$), $A_4\times D_6$ ($C_2^2$), $C_6\times S_4$ ($C_2^2$), $A_4\times C_3:C_4$ ($C_2^2$), $A_4:C_{12}$ ($C_2^2$)

Order 96: $C_2^3:D_6$ ($S_3$), $\GL(2,\mathbb{Z}/4)$ ($S_3$)

Order 72: $C_3:S_4$ ($D_4$) x 2, $C_6\times A_4$ ($C_2^3$)

Order 48: $C_2^3\times C_6$ ($D_6$), $C_2^2\times D_6$ ($D_6$), $C_2^2\times A_4$ ($D_6$), $C_2\times S_4$ ($D_6$), $C_6.C_2^3$ ($D_6$), $A_4:C_4$ ($D_6$)

Order 36: $C_3\times A_4$ ($C_2\times D_4$)

Order 24: $C_3:D_4$ ($S_4$), $C_2^2\times C_6$ ($C_2\times D_6$), $C_2\times A_4$ ($C_2\times D_6$)

Order 16: $C_2^4$ ($S_3^2$)

Order 12: $C_2\times C_6$ ($C_2\times S_4$), $C_2\times C_6$ ($S_3\times D_4$), $D_6$ ($C_2\times S_4$), $A_4$ ($S_3\times D_4$), $C_3:C_4$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($S_3\times D_6$)

Order 6: $C_6$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($S_3\times S_4$), $C_2^2$ ($D_6:D_6$)

Order 3: $C_3$ ($D_4\times S_4$)

Order 2: $C_2$ ($D_6\times S_4$)

Order 1: $C_1$ ($\GL(2,\mathbb{Z}/4):S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $\GL(2,\mathbb{Z}/4):S_3$ ($C_1$)

Order 288: $A_4\times C_3:D_4$ ($C_2$), $C_3\times \GL(2,\mathbb{Z}/4)$ ($C_2$), $D_6:S_4$ ($C_2$), $C_2^3.S_3^2$ ($C_2$), $C_2^3.S_3^2$ ($C_2$), $C_2\times C_6:S_4$ ($C_2$), $D_6\times S_4$ ($C_2$)

Order 144: $C_6:S_4$ ($C_2^2$) x 2, $C_2^2:C_6^2$ ($C_2^2$), $A_4\times D_6$ ($C_2^2$), $C_6\times S_4$ ($C_2^2$), $A_4\times C_3:C_4$ ($C_2^2$), $A_4:C_{12}$ ($C_2^2$)

Order 96: $C_2^3:D_6$ ($S_3$), $\GL(2,\mathbb{Z}/4)$ ($S_3$)

Order 72: $C_6\times A_4$ ($C_2^3$), $C_3:S_4$ ($D_4$)

Order 48: $C_2^3\times C_6$ ($D_6$), $C_2^2\times D_6$ ($D_6$), $C_2^2\times A_4$ ($D_6$), $C_2\times S_4$ ($D_6$), $C_6.C_2^3$ ($D_6$), $A_4:C_4$ ($D_6$)

Order 36: $C_3\times A_4$ ($C_2\times D_4$)

Order 24: $C_3:D_4$ ($S_4$), $C_2^2\times C_6$ ($C_2\times D_6$), $C_2\times A_4$ ($C_2\times D_6$)

Order 16: $C_2^4$ ($S_3^2$)

Order 12: $C_2\times C_6$ ($C_2\times S_4$), $C_2\times C_6$ ($S_3\times D_4$), $D_6$ ($C_2\times S_4$), $A_4$ ($S_3\times D_4$), $C_3:C_4$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($S_3\times D_6$)

Order 6: $C_6$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($S_3\times S_4$), $C_2^2$ ($D_6:D_6$)

Order 3: $C_3$ ($D_4\times S_4$)

Order 2: $C_2$ ($D_6\times S_4$)

Order 1: $C_1$ ($\GL(2,\mathbb{Z}/4):S_3$)

Series

Derived series

$\GL(2,\mathbb{Z}/4):S_3$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$\GL(2,\mathbb{Z}/4):S_3$

$\rhd$

$C_3\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_6\times S_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\GL(2,\mathbb{Z}/4):S_3$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 49 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $39 \times 39$ rational character table. Alternatively, you may search for characters of this group with desired properties.