Abstract group 128.1741: $(C_2\times Q_{16}):C_2^2$

• Label: 128.1741
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $20$
• Trans deg.: $32$
• Rank: $4$

Group information

Description:	$(C_2\times Q_{16}):C_2^2$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^7.C_2^5$, of order \(4096\)\(\medspace = 2^{12} \)
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	39	56	32	128
Conjugacy classes	1	13	12	6	32
Divisions	1	13	12	4	30
Autjugacy classes	1	6	7	2	16

Dimension	1	2	4	8
Irr. complex chars.	16	12	4	0	32
Irr. rational chars.	16	12	0	2	30

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$32$
Rank:	$4$
Inequivalent generating quadruples:	$20160$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{8}=[a,c]=[a,d]=[a,e]=[b,c]=[b,d]=[c,d]=[d,e]=1, b^{a}=be^{4}, e^{b}=de^{5}, e^{c}=e^{3} \rangle$
Permutation group:	Degree $20$ $\langle(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(18,20), (2,6)(4,12)(7,13)(8,9) \!\cdots\! \rangle$
Transitive group:	32T1385	32T1479			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_4.D_4)$ $\,\rtimes\,$ $C_2$	$(Q_8:C_4)$ $\,\rtimes\,$ $C_2^2$	$(C_2^3.D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^3.D_4)$ $\,\rtimes\,$ $C_2$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times Q_8)$ . $D_4$	$Q_8$ . $(C_2\times D_4)$	$(C_2\times D_4)$ . $D_4$ (3)	$(D_4:C_2)$ . $D_4$	all 22

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 796 subgroups in 376 conjugacy classes, 108 normal (28 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times D_4$
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:	$\operatorname{Fit} \simeq$ $(C_2\times Q_{16}):C_2^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2\times Q_{16}):C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2\times Q_{16}):C_2^2$

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

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Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 128: $(C_2\times Q_{16}):C_2^2$

Order 64: $D_4.D_4$ x 4, $C_2^2:\SD_{16}$ x 4, $C_2^3.D_4$, $C_2^3.D_4$, $(C_2^2\times C_8):C_2$, $D_4:C_2^3$, $Q_{16}:C_2^2$, $C_2^2\times \SD_{16}$, $C_4^2.C_2^2$

Order 32: $C_2\times \SD_{16}$ x 10, $D_4:C_2^2$ x 8, $D_4:C_4$ x 6, $C_2^2\times D_4$ x 6, $D_4:C_2^2$ x 5, $C_2^2:C_8$ x 4, $Q_{16}:C_2$ x 4, $C_2^2:Q_8$ x 4, $C_2\times Q_{16}$ x 2, $C_2^2.D_4$ x 2, $Q_8:C_4$ x 2, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4:Q_8$, $C_4^2:C_2$, $C_4^2:C_2$

Order 16: $C_2\times D_4$ x 50, $D_4:C_2$ x 26, $\SD_{16}$ x 12, $C_2\times Q_8$ x 9, $C_2^2\times C_4$ x 7, $C_2\times C_8$ x 6, $C_4:C_4$ x 6, $C_2^2:C_4$ x 5, $Q_{16}$ x 4, $C_2^4$ x 3, $\OD_{16}$ x 2, $C_4^2$

Order 8: $D_4$ x 39, $C_2\times C_4$ x 31, $C_2^3$ x 24, $Q_8$ x 11, $C_8$ x 4

Order 4: $C_2^2$ x 35, $C_4$ x 12

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $(C_2\times Q_{16}):C_2^2$

Order 64: $C_2^3.D_4$, $C_2^3.D_4$, $(C_2^2\times C_8):C_2$, $D_4:C_2^3$, $Q_{16}:C_2^2$, $C_2^2\times \SD_{16}$, $C_4^2.C_2^2$, $D_4.D_4$, $C_2^2:\SD_{16}$

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

Order 16: $C_2\times D_4$ x 14, $D_4:C_2$ x 7, $C_2\times Q_8$ x 6, $C_2^2\times C_4$ x 5, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_2^4$ x 2, $Q_{16}$, $\OD_{16}$, $C_4^2$

Order 8: $C_2\times C_4$ x 13, $D_4$ x 10, $C_2^3$ x 8, $Q_8$ x 6, $C_8$ x 2

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: $(C_2\times Q_{16}):C_2^2$ ($C_1$)

Order 64: $D_4.D_4$ ($C_2$) x 4, $C_2^2:\SD_{16}$ ($C_2$) x 4, $C_2^3.D_4$ ($C_2$), $C_2^3.D_4$ ($C_2$), $(C_2^2\times C_8):C_2$ ($C_2$), $D_4:C_2^3$ ($C_2$), $Q_{16}:C_2^2$ ($C_2$), $C_2^2\times \SD_{16}$ ($C_2$), $C_4^2.C_2^2$ ($C_2$)

Order 32: $D_4:C_4$ ($C_2^2$) x 6, $C_2\times \SD_{16}$ ($C_2^2$) x 6, $C_2^2:C_8$ ($C_2^2$) x 4, $D_4:C_2^2$ ($C_2^2$) x 4, $C_2^2:Q_8$ ($C_2^2$) x 4, $C_2^2\times D_4$ ($C_2^2$) x 3, $C_2\times Q_{16}$ ($C_2^2$) x 2, $Q_8:C_4$ ($C_2^2$) x 2, $C_2^2\times Q_8$ ($C_2^2$), $C_2\times \OD_{16}$ ($C_2^2$), $C_2^2\times C_8$ ($C_2^2$), $C_4^2:C_2$ ($C_2^2$)

Order 16: $C_2\times D_4$ ($D_4$) x 7, $C_2\times C_8$ ($C_2^3$) x 4, $D_4:C_2$ ($D_4$) x 4, $C_2\times Q_8$ ($C_2^3$) x 3, $C_2\times D_4$ ($C_2^3$) x 3, $C_2^2\times C_4$ ($C_2^3$) x 3, $C_4:C_4$ ($C_2^3$) x 2, $C_2\times Q_8$ ($D_4$)

Order 8: $C_2\times C_4$ ($C_2\times D_4$) x 7, $D_4$ ($C_2\times D_4$) x 6, $C_2^3$ ($C_2\times D_4$) x 3, $Q_8$ ($C_2\times D_4$) x 2, $C_2\times C_4$ ($C_2^4$)

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

Order 2: $C_2$ ($D_8:C_2^2$) x 2, $C_2$ ($C_2^3:D_4$)

Order 1: $C_1$ ($(C_2\times Q_{16}):C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $(C_2\times Q_{16}):C_2^2$ ($C_1$)

Order 64: $C_2^3.D_4$ ($C_2$), $C_2^3.D_4$ ($C_2$), $(C_2^2\times C_8):C_2$ ($C_2$), $D_4:C_2^3$ ($C_2$), $Q_{16}:C_2^2$ ($C_2$), $C_2^2\times \SD_{16}$ ($C_2$), $C_4^2.C_2^2$ ($C_2$), $D_4.D_4$ ($C_2$), $C_2^2:\SD_{16}$ ($C_2$)

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

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

Order 8: $C_2\times C_4$ ($C_2\times D_4$) x 4, $C_2^3$ ($C_2\times D_4$) x 2, $D_4$ ($C_2\times D_4$) x 2, $Q_8$ ($C_2\times D_4$), $C_2\times C_4$ ($C_2^4$)

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

Order 2: $C_2$ ($D_8:C_2^2$), $C_2$ ($C_2^3:D_4$)

Order 1: $C_1$ ($(C_2\times Q_{16}):C_2^2$)

Series

Derived series

$(C_2\times Q_{16}):C_2^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$(C_2\times Q_{16}):C_2^2$

$\rhd$

$C_2^2:\SD_{16}$

$\rhd$

$C_2\times \SD_{16}$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2\times Q_{16}):C_2^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$D_4:C_2^2$

$\lhd$

$(C_2\times Q_{16}):C_2^2$

Supergroups

This group is a maximal subgroup of 100 larger groups in the database.

This group is a maximal quotient of 214 larger groups in the database.

Character theory

Complex character table

See the $32 \times 32$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.