Abstract group 128.854: $(C_2^2\times D_4):C_4$

• Label: 128.854
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$(C_2^2\times D_4):C_4$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^4.C_2^6$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$4$
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	31	64	32	128
Conjugacy classes	1	7	11	4	23
Divisions	1	7	9	2	19
Autjugacy classes	1	6	7	1	15

Dimension	1	2	4	8
Irr. complex chars.	16	4	2	1	23
Irr. rational chars.	8	8	2	1	19

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$672$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	8	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{2}=c^{4}=d^{4}=[a,d]=[b,d]=1, b^{a}=bc^{2}, c^{a}=bcd^{2}, c^{b}=cd^{2}, d^{c}=d^{3} \rangle$
Permutation group:	Degree $16$ $\langle(1,9)(2,10)(3,12,4,11)(5,13)(6,14)(7,16,8,15), (1,5)(2,6)(3,7)(4,8)(9,15) \!\cdots\! \rangle$
Transitive group:	16T213	16T281	16T315	32T466	all 11
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\wr C_4)$ $\,\rtimes\,$ $C_2$ (4)	$(C_2^2\times D_4)$ $\,\rtimes\,$ $C_4$	$(C_4^2:C_2)$ $\,\rtimes\,$ $C_4$	$(C_2^2\wr C_2)$ $\,\rtimes\,$ $C_4$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times Q_8)$ . $D_4$	$(C_2\times D_4)$ . $D_4$	$(C_2\times D_4)$ . $C_2^3$	$(C_2^2\times C_4)$ . $D_4$ (2)	all 21

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 380 subgroups in 133 conjugacy classes, 42 normal (26 characteristic).

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

Special subgroups

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

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 128: $(C_2^2\times D_4):C_4$

Order 64: $C_2\wr C_4$ x 4, $\OD_{16}.C_2^2$, $C_2^3.C_2^3$, $C_4^2:C_2^2$

Order 32: $C_2^2\wr C_2$ x 4, $C_2^3:C_4$ x 3, $\OD_{16}:C_2$ x 2, $C_4:D_4$ x 2, $C_4^2:C_2$ x 2, $C_4.D_4$, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \OD_{16}$, $C_4:D_4$, $C_4^2:C_2$

Order 16: $C_2\times D_4$ x 11, $C_2^2:C_4$ x 8, $\OD_{16}$ x 3, $C_2^2\times C_4$ x 3, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2^4$ x 2, $D_4:C_2$ x 2, $C_2\times C_8$, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 14, $D_4$ x 11, $C_2^3$ x 9, $C_8$ x 2, $Q_8$

Order 4: $C_2^2$ x 17, $C_4$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $(C_2^2\times D_4):C_4$

Order 64: $\OD_{16}.C_2^2$, $C_2^3.C_2^3$, $C_2\wr C_4$, $C_4^2:C_2^2$

Order 32: $C_2^3:C_4$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_4^2:C_2$ x 2, $C_4.D_4$, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2\times \OD_{16}$, $C_4:D_4$, $C_4^2:C_2$

Order 16: $C_2\times D_4$ x 9, $C_2^2:C_4$ x 4, $C_2^2\times C_4$ x 3, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_4^2$ x 2, $D_4:C_2$ x 2, $C_2\times C_8$, $C_2^4$, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 12, $D_4$ x 8, $C_2^3$ x 6, $Q_8$, $C_8$

Order 4: $C_2^2$ x 11, $C_4$ x 7

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $(C_2^2\times D_4):C_4$ ($C_1$)

Order 64: $C_2\wr C_4$ ($C_2$) x 4, $\OD_{16}.C_2^2$ ($C_2$), $C_2^3.C_2^3$ ($C_2$), $C_4^2:C_2^2$ ($C_2$)

Order 32: $\OD_{16}:C_2$ ($C_2^2$) x 2, $C_2^3:C_4$ ($C_2^2$) x 2, $C_2^2\wr C_2$ ($C_2^2$) x 2, $C_2^2\wr C_2$ ($C_4$) x 2, $D_4:C_2^2$ ($C_2^2$), $C_2^2\times D_4$ ($C_4$), $C_4^2:C_2$ ($C_4$)

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

Order 8: $C_2\times C_4$ ($C_2^2:C_4$) x 3, $C_2^3$ ($C_2^2:C_4$), $C_2^3$ ($C_2\times D_4$), $C_2^3$ ($C_2^2\times C_4$), $C_2\times C_4$ ($C_2\times D_4$)

Order 4: $C_4$ ($C_2^3:C_4$) x 2, $C_2^2$ ($C_2^3:C_4$)

Order 2: $C_2$ ($C_2^4:C_4$)

Order 1: $C_1$ ($(C_2^2\times D_4):C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $(C_2^2\times D_4):C_4$ ($C_1$)

Order 64: $\OD_{16}.C_2^2$ ($C_2$), $C_2^3.C_2^3$ ($C_2$), $C_2\wr C_4$ ($C_2$), $C_4^2:C_2^2$ ($C_2$)

Order 32: $\OD_{16}:C_2$ ($C_2^2$), $C_2^3:C_4$ ($C_2^2$), $D_4:C_2^2$ ($C_2^2$), $C_2^2\times D_4$ ($C_4$), $C_2^2\wr C_2$ ($C_2^2$), $C_2^2\wr C_2$ ($C_4$), $C_4^2:C_2$ ($C_4$)

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

Order 8: $C_2\times C_4$ ($C_2^2:C_4$) x 3, $C_2^3$ ($C_2^2:C_4$), $C_2^3$ ($C_2\times D_4$), $C_2^3$ ($C_2^2\times C_4$), $C_2\times C_4$ ($C_2\times D_4$)

Order 4: $C_4$ ($C_2^3:C_4$) x 2, $C_2^2$ ($C_2^3:C_4$)

Order 2: $C_2$ ($C_2^4:C_4$)

Order 1: $C_1$ ($(C_2^2\times D_4):C_4$)

Series

Derived series

$(C_2^2\times D_4):C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$(C_2^2\times D_4):C_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times D_4):C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_4:C_2^2$

$\lhd$

$(C_2^2\times D_4):C_4$

Supergroups

This group is a maximal subgroup of 39 larger groups in the database.

This group is a maximal quotient of 58 larger groups in the database.

Character theory

Complex character table

See the $23 \times 23$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.