Abstract group 768.1088320: $D_4.\GL(2,\mathbb{Z}/4)$

• Label: 768.1088320
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $16$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$D_4.\GL(2,\mathbb{Z}/4)$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$A_4.C_2^6.C_2^4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	95	8	224	184	192	64	768
Conjugacy classes	1	19	1	10	15	8	4	58
Divisions	1	19	1	10	9	4	3	47
Autjugacy classes	1	11	1	8	5	2	2	30

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	8	26	8	5	10	0	1	58
Irr. rational chars.	8	10	8	11	6	1	3	47

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$10080$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{2}=c^{8}=d^{6}=e^{2}=f^{2}=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,4,7)(3,6,8,5)(9,10,11,12)(14,16), (1,3,4,8)(2,5,7,6)(10,12), (2,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 10 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 5 & 6 \\ 6 & 11 \end{array}\right), \left(\begin{array}{rr} 5 & 10 \\ 1 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 6 \\ 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^4.D_6)$ $\,\rtimes\,$ $C_2^2$	$(C_2^2\times A_4)$ $\,\rtimes\,$ $\SD_{16}$	$A_4$ $\,\rtimes\,$ $(C_2^2:\SD_{16})$	$C_2^2$ $\,\rtimes\,$ $(A_4:\SD_{16})$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4\times A_4)$ . $D_4$	$(D_4\times C_2^4)$ . $S_3$	$(C_2^2\times D_4)$ . $S_4$	$(C_2^3\times A_4)$ . $D_4$	all 36

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{12}\Z)$.

Homology

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

Subgroups

There are 10162 subgroups in 1479 conjugacy classes, 71 normal (35 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times \GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $C_2^3:C_{12}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_2^4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $D_4.\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_6:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 768: $D_4.\GL(2,\mathbb{Z}/4)$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^6:C_6$, $C_4.\GL(2,\mathbb{Z}/4)$

Order 256: $C_2^4:\SD_{16}$

Order 192: $C_2^5:C_6$ x 7, $A_4:\SD_{16}$ x 4, $C_2\times A_4:C_8$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2^3.D_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6):\SD_{16}$, $A_4\times C_2^4$, $C_2^4:C_{12}$, $C_2^4.D_6$

Order 128: $C_2^3.D_8$ x 6, $C_2^3:\SD_{16}$ x 6, $C_2^4:C_8$, $D_4\times C_2^4$, $C_2^4:Q_8$

Order 96: $C_2^3\times A_4$ x 11, $D_4\times A_4$ x 10, $C_2^3:C_{12}$ x 4, $A_4:C_8$ x 2, $C_6.D_8$ x 2, $C_2^2.S_4$ x 2, $A_4:Q_8$ x 2, $C_6:\SD_{16}$ x 2, $C_{12}.D_4$, $C_{12}:C_2^3$, $C_{12}.D_4$

Order 64: $C_2^2:\SD_{16}$ x 24, $D_4\times C_2^3$ x 17, $C_2^2.D_8$ x 12, $C_2^3:C_8$ x 6, $C_2^2\times \SD_{16}$ x 4, $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^3:Q_8$ x 3, $C_2^4:C_4$, $C_2^6$, $C_2^4\times C_4$

Order 48: $C_2^2\times A_4$ x 21, $C_6\times D_4$ x 7, $C_3:\SD_{16}$ x 4, $C_4\times A_4$ x 3, $C_6:C_8$ x 2, $A_4:C_4$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6:Q_8$, $C_6.D_4$, $C_{12}:C_4$, $C_6.D_4$

Order 32: $C_2^2\times D_4$ x 113, $D_4:C_4$ x 24, $C_2\times \SD_{16}$ x 24, $C_2^5$ x 23, $C_2^2:C_8$ x 12, $C_2^2:Q_8$ x 12, $C_2^3\times C_4$ x 10, $C_2^2.D_4$ x 6, $C_2^3:C_4$ x 6, $C_2^2\times C_8$ x 4, $C_2.C_4^2$ x 3, $C_2^2\times Q_8$

Order 24: $C_2^2\times C_6$ x 11, $C_3\times D_4$ x 10, $C_2\times A_4$ x 9, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2

Order 16: $C_2\times D_4$ x 176, $C_2^4$ x 149, $C_2^2\times C_4$ x 37, $\SD_{16}$ x 16, $C_2\times C_8$ x 12, $C_4:C_4$ x 12, $C_2^2:C_4$ x 12, $C_2\times Q_8$ x 6

Order 12: $C_2\times C_6$ x 21, $C_{12}$ x 3, $C_3:C_4$ x 2, $A_4$

Order 8: $C_2^3$ x 274, $D_4$ x 56, $C_2\times C_4$ x 40, $Q_8$ x 4, $C_8$ x 4

Order 6: $C_6$ x 9

Order 4: $C_2^2$ x 134, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $D_4.\GL(2,\mathbb{Z}/4)$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^6:C_6$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$

Order 256: $C_2^4:\SD_{16}$

Order 192: $C_2^5:C_6$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:\SD_{16}$, $C_2^3.D_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2\times A_4:C_8$, $(C_2\times C_6):\SD_{16}$, $A_4\times C_2^4$, $C_2^4:C_{12}$, $C_2^4.D_6$

Order 128: $C_2^3.D_8$ x 4, $C_2^3:\SD_{16}$ x 4, $C_2^4:C_8$, $D_4\times C_2^4$, $C_2^4:Q_8$

Order 96: $C_2^3\times A_4$ x 5, $D_4\times A_4$ x 3, $C_2^3:C_{12}$ x 3, $C_2^2.S_4$ x 2, $A_4:C_8$, $C_6.D_8$, $C_{12}.D_4$, $C_{12}:C_2^3$, $A_4:Q_8$, $C_6:\SD_{16}$, $C_{12}.D_4$

Order 64: $D_4\times C_2^3$ x 9, $C_2^2:\SD_{16}$ x 8, $C_2^2.D_8$ x 6, $C_2^3:C_8$ x 4, $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^3:Q_8$ x 3, $C_2^2\times \SD_{16}$ x 2, $C_2^4:C_4$, $C_2^6$, $C_2^4\times C_4$

Order 48: $C_2^2\times A_4$ x 8, $C_6\times D_4$ x 3, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_6:C_8$, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6:Q_8$, $C_6.D_4$, $C_3:\SD_{16}$, $C_{12}:C_4$, $C_6.D_4$

Order 32: $C_2^2\times D_4$ x 33, $C_2^5$ x 11, $C_2^3\times C_4$ x 8, $D_4:C_4$ x 7, $C_2^2:C_8$ x 7, $C_2\times \SD_{16}$ x 7, $C_2^2:Q_8$ x 7, $C_2^2.D_4$ x 6, $C_2^3:C_4$ x 6, $C_2.C_4^2$ x 3, $C_2^2\times C_8$ x 2, $C_2^2\times Q_8$

Order 24: $C_2^2\times C_6$ x 5, $C_2\times A_4$ x 5, $C_2\times C_{12}$ x 3, $C_3\times D_4$ x 3, $C_6:C_4$ x 2, $C_3:Q_8$, $C_3:C_8$

Order 16: $C_2^4$ x 44, $C_2\times D_4$ x 36, $C_2^2\times C_4$ x 25, $C_2^2:C_4$ x 9, $C_4:C_4$ x 8, $C_2\times C_8$ x 6, $C_2\times Q_8$ x 4, $\SD_{16}$ x 3

Order 12: $C_2\times C_6$ x 8, $C_{12}$ x 2, $C_3:C_4$ x 2, $A_4$

Order 8: $C_2^3$ x 69, $C_2\times C_4$ x 29, $D_4$ x 12, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 44, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $D_4.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^6:C_6$ ($C_2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$)

Order 192: $C_2\times A_4:C_8$ ($C_2^2$) x 2, $C_2^5:C_6$ ($C_2^2$) x 2, $C_2^3.D_{12}$ ($C_2^2$), $C_2^4:C_{12}$ ($C_2^2$), $C_2^4.D_6$ ($C_2^2$)

Order 128: $D_4\times C_2^4$ ($S_3$)

Order 96: $D_4\times A_4$ ($D_4$) x 4, $C_2^3\times A_4$ ($D_4$), $C_2^3:C_{12}$ ($C_2^3$), $C_2^3:C_{12}$ ($D_4$)

Order 64: $D_4\times C_2^3$ ($D_6$) x 2, $C_2^4\times C_4$ ($D_6$)

Order 48: $C_2^2\times A_4$ ($\SD_{16}$) x 2, $C_4\times A_4$ ($C_2\times D_4$) x 2, $C_2^2\times A_4$ ($C_2\times D_4$)

Order 32: $C_2^2\times D_4$ ($C_3:D_4$) x 4, $C_2^5$ ($C_3:D_4$), $C_2^2\times D_4$ ($S_4$), $C_2^3\times C_4$ ($C_3:D_4$), $C_2^3\times C_4$ ($C_2\times D_6$)

Order 24: $C_2\times A_4$ ($D_8:C_2$), $C_2\times A_4$ ($C_2\times \SD_{16}$), $C_2\times A_4$ ($C_2^2\wr C_2$)

Order 16: $C_2^4$ ($C_3:\SD_{16}$) x 2, $C_2\times D_4$ ($C_2\times S_4$) x 2, $C_2^2\times C_4$ ($C_6:D_4$) x 2, $C_2^4$ ($C_6:D_4$), $C_2^2\times C_4$ ($C_2\times S_4$)

Order 12: $A_4$ ($C_2^2:\SD_{16}$)

Order 8: $D_4$ ($\GL(2,\mathbb{Z}/4)$) x 4, $C_2^3$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3$ ($C_2^4:S_3$), $C_2^3$ ($C_6:\SD_{16}$), $C_2^3$ ($D_{12}:C_2^2$), $C_2\times C_4$ ($C_2^2\times S_4$), $C_2\times C_4$ ($\GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($A_4:\SD_{16}$) x 2, $C_4$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 2, $C_2^2$ ($(C_2\times C_6):\SD_{16}$), $C_2^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^2:\GL(2,\mathbb{Z}/4)$), $C_2$ ($C_4.\GL(2,\mathbb{Z}/4)$), $C_2$ ($C_4.\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($D_4.\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $D_4.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^6:C_6$ ($C_2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2$)

Order 192: $C_2^3.D_{12}$ ($C_2^2$), $C_2\times A_4:C_8$ ($C_2^2$), $C_2^5:C_6$ ($C_2^2$), $C_2^4:C_{12}$ ($C_2^2$), $C_2^4.D_6$ ($C_2^2$)

Order 128: $D_4\times C_2^4$ ($S_3$)

Order 96: $C_2^3\times A_4$ ($D_4$), $D_4\times A_4$ ($D_4$), $C_2^3:C_{12}$ ($C_2^3$), $C_2^3:C_{12}$ ($D_4$)

Order 64: $D_4\times C_2^3$ ($D_6$), $C_2^4\times C_4$ ($D_6$)

Order 48: $C_2^2\times A_4$ ($\SD_{16}$), $C_2^2\times A_4$ ($C_2\times D_4$), $C_4\times A_4$ ($C_2\times D_4$)

Order 32: $C_2^5$ ($C_3:D_4$), $C_2^2\times D_4$ ($C_3:D_4$), $C_2^2\times D_4$ ($S_4$), $C_2^3\times C_4$ ($C_3:D_4$), $C_2^3\times C_4$ ($C_2\times D_6$)

Order 24: $C_2\times A_4$ ($D_8:C_2$), $C_2\times A_4$ ($C_2\times \SD_{16}$), $C_2\times A_4$ ($C_2^2\wr C_2$)

Order 16: $C_2^4$ ($C_6:D_4$), $C_2^4$ ($C_3:\SD_{16}$), $C_2\times D_4$ ($C_2\times S_4$), $C_2^2\times C_4$ ($C_2\times S_4$), $C_2^2\times C_4$ ($C_6:D_4$)

Order 12: $A_4$ ($C_2^2:\SD_{16}$)

Order 8: $C_2^3$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3$ ($C_2^4:S_3$), $C_2^3$ ($C_6:\SD_{16}$), $C_2^3$ ($D_{12}:C_2^2$), $D_4$ ($\GL(2,\mathbb{Z}/4)$), $C_2\times C_4$ ($C_2^2\times S_4$), $C_2\times C_4$ ($\GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($A_4:\SD_{16}$), $C_2^2$ ($(C_2\times C_6):\SD_{16}$), $C_2^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_4$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^2:\GL(2,\mathbb{Z}/4)$), $C_2$ ($C_4.\GL(2,\mathbb{Z}/4)$), $C_2$ ($C_4.\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($D_4.\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$D_4.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^3:C_{12}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$D_4.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^6:C_6$

$\rhd$

$C_2^4:C_{12}$

$\rhd$

$C_2^3:C_{12}$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$D_4.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^3:C_{12}$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2\times D_4$

Supergroups

This group is a maximal subgroup of 1 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 $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $47 \times 47$ rational character table.