Abstract group 128.1762: $C_4^2.D_4$

• Label: 128.1762
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $14$
• Trans deg.: $64$
• Rank: $4$

Group information

Description:	$C_4^2.D_4$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^5.(D_4\times S_4)$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4	8
Elements	1	23	72	32	128
Conjugacy classes	1	11	18	8	38
Divisions	1	11	16	4	32
Autjugacy classes	1	5	7	1	14

Dimension	1	2	4
Irr. complex chars.	16	20	2	38
Irr. rational chars.	16	8	8	32

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$64$
Rank:	$4$
Inequivalent generating quadruples:	$5040$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{8}=d^{4}=[a,b]=[a,c]=[a,d]=[b,d]=1, c^{b}=c^{3}, d^{c}=d^{3} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,4)(5,6)(7,8,10,13)(9,11,14,12), (1,3,4,2), (1,4)(2,3)(8,11)(9,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 9 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 10 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 9 & 8 \\ 4 & 9 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 9 \\ 6 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 2 \\ 2 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 2 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_4:\SD_{16})$
Semidirect product:	$(C_4:Q_8)$ $\,\rtimes\,$ $C_2^2$	$(Q_8:C_4)$ $\,\rtimes\,$ $C_2^2$	$(C_4^2.C_4)$ $\,\rtimes\,$ $C_2$	$(C_4:C_8)$ $\,\rtimes\,$ $C_2^2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $D_4$	$C_4^2$ . $C_2^3$	$(C_2\times D_4)$ . $D_4$	$D_4$ . $(C_2\times D_4)$	all 33

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 524 subgroups in 272 conjugacy classes, 116 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2\times D_4$
Commutator:	$G' \simeq$ $C_2\times C_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^2.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.D_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 128: $C_4^2.D_4$

Order 64: $C_4:\SD_{16}$ x 8, $C_2^2.Q_{16}$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_4$

Order 32: $C_2\times \SD_{16}$ x 16, $Q_8:C_4$ x 8, $C_4:Q_8$ x 6, $C_4\times D_4$ x 6, $C_4:C_8$ x 4, $C_2^2.D_4$ x 3, $C_2^2\times Q_8$ x 2, $C_2^2\times C_8$ x 2, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_2^3:C_4$, $C_2\times C_4^2$

Order 16: $\SD_{16}$ x 16, $C_2\times Q_8$ x 14, $C_2^2\times C_4$ x 14, $C_4:C_4$ x 11, $C_2\times D_4$ x 9, $C_2\times C_8$ x 8, $C_2^2:C_4$ x 4, $C_4^2$ x 4, $C_2^4$

Order 8: $C_2\times C_4$ x 36, $Q_8$ x 12, $C_2^3$ x 11, $D_4$ x 10, $C_8$ x 4

Order 4: $C_2^2$ x 23, $C_4$ x 16

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_4^2.D_4$

Order 64: $C_2^2.Q_{16}$, $C_2^2\times \SD_{16}$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4:\SD_{16}$, $C_4^2.C_4$

Order 32: $C_2^2.D_4$ x 3, $C_2\times \SD_{16}$ x 2, $C_4:Q_8$ x 2, $C_4\times D_4$ x 2, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_2^2\times C_8$, $C_2^3:C_4$, $C_2\times C_4^2$, $C_4:C_8$, $Q_8:C_4$

Order 16: $C_2^2\times C_4$ x 7, $C_4:C_4$ x 4, $C_2\times Q_8$ x 4, $C_2\times D_4$ x 4, $C_2\times C_8$ x 2, $C_4^2$ x 2, $\SD_{16}$, $C_2^2:C_4$, $C_2^4$

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 5, $Q_8$ x 3, $D_4$ x 3, $C_8$

Order 4: $C_2^2$ x 8, $C_4$ x 7

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_4^2.D_4$ ($C_1$)

Order 64: $C_4:\SD_{16}$ ($C_2$) x 8, $C_2^2.Q_{16}$ ($C_2$) x 2, $C_2^2\times \SD_{16}$ ($C_2$) x 2, $C_4^2.C_2^2$ ($C_2$), $C_4^2:C_2^2$ ($C_2$), $C_4^2.C_4$ ($C_2$)

Order 32: $C_2\times \SD_{16}$ ($C_2^2$) x 8, $Q_8:C_4$ ($C_2^2$) x 8, $C_4:Q_8$ ($C_2^2$) x 4, $C_4\times D_4$ ($C_2^2$) x 4, $C_4:C_8$ ($C_2^2$) x 4, $C_2^2\times Q_8$ ($C_2^2$) x 2, $C_2^2\times C_8$ ($C_2^2$) x 2, $C_2^2\times D_4$ ($C_2^2$), $C_2^2.D_4$ ($C_2^2$), $C_2\times C_4^2$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($C_2^3$) x 4, $C_2\times Q_8$ ($C_2^3$) x 4, $C_2\times D_4$ ($D_4$) x 4, $C_4:C_4$ ($C_2^3$) x 2, $C_4^2$ ($C_2^3$) x 2, $C_4^2$ ($D_4$) x 2, $C_2\times D_4$ ($C_2^3$) x 2, $C_2^2\times C_4$ ($D_4$) x 2, $C_2^2\times C_4$ ($C_2^3$)

Order 8: $C_2\times C_4$ ($C_2\times D_4$) x 7, $D_4$ ($C_2\times D_4$) x 4, $C_2\times C_4$ ($\SD_{16}$) x 4, $C_2\times C_4$ ($D_4:C_2$) x 2, $C_2^3$ ($C_2\times D_4$), $C_2\times C_4$ ($C_2^4$)

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

Order 2: $C_2$ ($C_4:\SD_{16}$) x 4, $C_2$ ($Q_{16}:C_2^2$), $C_2$ ($C_2^2\times \SD_{16}$), $C_2$ ($C_2^3:D_4$)

Order 1: $C_1$ ($C_4^2.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_4^2.D_4$ ($C_1$)

Order 64: $C_2^2.Q_{16}$ ($C_2$), $C_2^2\times \SD_{16}$ ($C_2$), $C_4^2.C_2^2$ ($C_2$), $C_4^2:C_2^2$ ($C_2$), $C_4:\SD_{16}$ ($C_2$), $C_4^2.C_4$ ($C_2$)

Order 32: $C_2^2\times Q_8$ ($C_2^2$), $C_2^2\times D_4$ ($C_2^2$), $C_2\times \SD_{16}$ ($C_2^2$), $C_2^2\times C_8$ ($C_2^2$), $C_4:Q_8$ ($C_2^2$), $C_4\times D_4$ ($C_2^2$), $C_2^2.D_4$ ($C_2^2$), $C_2\times C_4^2$ ($C_2^2$), $C_4:C_8$ ($C_2^2$), $Q_8:C_4$ ($C_2^2$)

Order 16: $C_2^2\times C_4$ ($D_4$) x 2, $C_2\times C_8$ ($C_2^3$), $C_4:C_4$ ($C_2^3$), $C_4^2$ ($C_2^3$), $C_4^2$ ($D_4$), $C_2\times Q_8$ ($C_2^3$), $C_2\times D_4$ ($C_2^3$), $C_2\times D_4$ ($D_4$), $C_2^2\times C_4$ ($C_2^3$)

Order 8: $C_2\times C_4$ ($C_2\times D_4$) x 4, $C_2^3$ ($C_2\times D_4$), $D_4$ ($C_2\times D_4$), $C_2\times C_4$ ($\SD_{16}$), $C_2\times C_4$ ($C_2^4$), $C_2\times C_4$ ($D_4:C_2$)

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

Order 2: $C_2$ ($Q_{16}:C_2^2$), $C_2$ ($C_2^2\times \SD_{16}$), $C_2$ ($C_2^3:D_4$), $C_2$ ($C_4:\SD_{16}$)

Order 1: $C_1$ ($C_4^2.D_4$)

Series

Derived series

$C_4^2.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$C_4^2.D_4$

$\rhd$

$C_4:\SD_{16}$

$\rhd$

$C_4:C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^2.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$C_4^2.D_4$

Supergroups

This group is a maximal subgroup of 307 larger groups in the database.

This group is a maximal quotient of 78 larger groups in the database.

Character theory

Complex character table

See the $38 \times 38$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.