Abstract group $\GL(2,\mathbb{Z}/4):C_2$

• Label: 192.1471
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$\GL(2,\mathbb{Z}/4):C_2$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\GL(2,\mathbb{Z}/4):C_2^2$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Outer automorphisms:	$C_2^3$, of order \(8\)\(\medspace = 2^{3} \)
Composition factors:	$C_2$ x 6, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	39	8	88	24	32	192
Conjugacy classes	1	7	1	12	3	4	28
Divisions	1	7	1	10	2	2	23
Autjugacy classes	1	6	1	7	2	2	19

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	8	10	8	0	2	0	0	28
Irr. rational chars.	8	4	8	1	0	1	1	23

Minimal Presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$5040$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	7	7

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{2}=d^{2}=e^{4}=[a,d]=[a,e]=[b,e]=[c,d]=[c,e]=[d,e]=1, b^{a}=b^{5}e^{2}, c^{a}=cd, c^{b}=d, d^{b}=cd \rangle$
Permutation group:	Degree $12$ $\langle(2,3)(5,6)(7,9)(8,10)(11,12), (5,7,8,11)(6,9,10,12), (6,10)(9,12), (5,8)(6,10)(7,11)(9,12), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 8 \\ 3 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 5 \\ 7 & 12 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 9 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 10 & 11 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 18 & 15 \\ 3 & 2 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 10 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Transitive group:	24T292	24T319	24T395	32T2182	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4\times S_4)$ $\,\rtimes\,$ $C_2$ (2)	$(C_4:S_4)$ $\,\rtimes\,$ $C_2$	$(C_2\times C_4)$ $\,\rtimes\,$ $S_4$	$(A_4:Q_8)$ $\,\rtimes\,$ $C_2$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $D_6$	$C_4$ . $(C_2\times S_4)$ (2)	$(C_2\times S_4)$ . $C_2^2$ (2)	$C_2^2$ . $(C_2\times S_4)$	all 11

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

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

Subgroups

There are 626 subgroups in 171 conjugacy classes, 29 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_2\times S_4$
Commutator:	$G' \simeq$ $C_2\times A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $\GL(2,\mathbb{Z}/4):C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 192: $\GL(2,\mathbb{Z}/4):C_2$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times S_4$ x 2, $C_2^3:C_{12}$, $C_4:S_4$, $A_4:Q_8$

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

Order 48: $C_2\times S_4$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2\times A_4$, $D_{12}:C_2$

Order 32: $C_4\times D_4$ x 4, $C_2^2.D_4$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $D_4:C_2^2$, $C_2^3\times C_4$, $C_4^2:C_2$

Order 24: $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_3:Q_8$

Order 16: $C_2^2:C_4$ x 10, $C_2^2\times C_4$ x 8, $C_2\times D_4$ x 7, $C_4:C_4$ x 6, $D_4:C_2$ x 4, $C_4^2$ x 2, $C_2^4$, $C_2\times Q_8$

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

Order 8: $C_2\times C_4$ x 19, $D_4$ x 14, $C_2^3$ x 7, $Q_8$ x 2

Order 6: $C_6$ x 2, $S_3$ x 2

Order 4: $C_2^2$ x 16, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $\GL(2,\mathbb{Z}/4):C_2$

Order 96: $C_2^3:C_{12}$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $C_4\times S_4$, $A_4:Q_8$

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

Order 48: $C_4\times A_4$ x 2, $C_2^2\times A_4$, $C_2\times S_4$, $D_{12}:C_2$, $A_4:C_4$

Order 32: $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $D_4:C_2^2$, $C_2^3\times C_4$, $C_2^2.D_4$, $C_2^2\wr C_2$, $C_4^2:C_2$

Order 24: $C_2\times A_4$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $S_4$

Order 16: $C_2^2\times C_4$ x 7, $C_2^2:C_4$ x 5, $C_4:C_4$ x 4, $D_4:C_2$ x 4, $C_2\times D_4$ x 4, $C_4^2$, $C_2^4$, $C_2\times Q_8$

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

Order 8: $C_2\times C_4$ x 14, $D_4$ x 7, $C_2^3$ x 6, $Q_8$ x 2

Order 6: $C_6$ x 2, $S_3$

Order 4: $C_2^2$ x 12, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $\GL(2,\mathbb{Z}/4):C_2$ ($C_1$)

Order 96: $\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $C_4\times S_4$ ($C_2$) x 2, $C_2^3:C_{12}$ ($C_2$), $C_4:S_4$ ($C_2$), $A_4:Q_8$ ($C_2$)

Order 48: $C_2\times S_4$ ($C_2^2$) x 2, $C_4\times A_4$ ($C_2^2$) x 2, $A_4:C_4$ ($C_2^2$) x 2, $C_2^2\times A_4$ ($C_2^2$)

Order 32: $C_2^3\times C_4$ ($S_3$)

Order 24: $C_2\times A_4$ ($C_2^3$)

Order 16: $C_2^2\times C_4$ ($D_6$) x 2, $C_2^4$ ($D_6$)

Order 12: $A_4$ ($D_4:C_2$)

Order 8: $C_2^3$ ($C_2\times D_6$), $C_2\times C_4$ ($S_4$)

Order 4: $C_4$ ($C_2\times S_4$) x 2, $C_2^2$ ($C_2\times S_4$), $C_2^2$ ($D_{12}:C_2$)

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

Order 1: $C_1$ ($\GL(2,\mathbb{Z}/4):C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $\GL(2,\mathbb{Z}/4):C_2$ ($C_1$)

Order 96: $C_2^3:C_{12}$ ($C_2$), $\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_4:S_4$ ($C_2$), $C_4\times S_4$ ($C_2$), $A_4:Q_8$ ($C_2$)

Order 48: $C_4\times A_4$ ($C_2^2$) x 2, $C_2^2\times A_4$ ($C_2^2$), $C_2\times S_4$ ($C_2^2$), $A_4:C_4$ ($C_2^2$)

Order 32: $C_2^3\times C_4$ ($S_3$)

Order 24: $C_2\times A_4$ ($C_2^3$)

Order 16: $C_2^2\times C_4$ ($D_6$) x 2, $C_2^4$ ($D_6$)

Order 12: $A_4$ ($D_4:C_2$)

Order 8: $C_2^3$ ($C_2\times D_6$), $C_2\times C_4$ ($S_4$)

Order 4: $C_4$ ($C_2\times S_4$) x 2, $C_2^2$ ($C_2\times S_4$), $C_2^2$ ($D_{12}:C_2$)

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

Order 1: $C_1$ ($\GL(2,\mathbb{Z}/4):C_2$)

Series

Derived series

$\GL(2,\mathbb{Z}/4):C_2$

$\rhd$

$C_2\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$\GL(2,\mathbb{Z}/4):C_2$

$\rhd$

$C_4\times S_4$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\GL(2,\mathbb{Z}/4):C_2$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 46 larger groups in the database.

This group is a maximal quotient of 69 larger groups in the database.

Character theory

Complex character table

See the $28 \times 28$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $23 \times 23$ rational character table.