Abstract group 128.163: $Q_{128}$

• Label: 128.163
• Order: \( 2^{7} \)
• Exponent: \( 2^{6} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $128$
• Trans deg.: $128$
• Rank: $2$

Group information

Description:	$Q_{128}$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(64\)\(\medspace = 2^{6} \)
Automorphism group:	$D_{64}:C_{16}$, of order \(2048\)\(\medspace = 2^{11} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$6$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	32	64
Elements	1	1	66	4	8	16	32	128
Conjugacy classes	1	1	3	2	4	8	16	35
Divisions	1	1	3	1	1	1	1	9
Autjugacy classes	1	1	2	1	1	1	1	8

Dimension	1	2	4	8	16	32
Irr. complex chars.	4	31	0	0	0	0	35
Irr. rational chars.	4	1	1	1	1	1	9

Minimal presentations

Permutation degree:	$128$
Transitive degree:	$128$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	64
Arbitrary	2	4	64

Constructions

Presentation:	$\langle a, b \mid b^{64}=1, a^{2}=b^{32}, b^{a}=b^{63} \rangle$
Permutation group:	Degree $128$ $\langle(1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 98 & 85 \\ 14 & 29 \end{array}\right), \left(\begin{array}{rr} 112 & 52 \\ 102 & 112 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{127})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $D_8$	$Q_{64}$ . $C_2$ (2)	$C_{16}$ . $D_4$	$C_4$ . $D_{16}$	all 7

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 70 subgroups in 18 conjugacy classes, 10 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{32}$
Commutator:	$G' \simeq$ $C_{32}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $Q_{128}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $Q_{128}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $D_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{128}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 128: $Q_{128}$

Order 64: $Q_{64}$ x 2, $C_{64}$

Order 32: $Q_{32}$ x 2, $C_{32}$

Order 16: $Q_{16}$ x 2, $C_{16}$

Order 8: $Q_8$ x 2, $C_8$

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $Q_{128}$

Order 64: $Q_{64}$, $C_{64}$

Order 32: $Q_{32}$, $C_{32}$

Order 16: $Q_{16}$, $C_{16}$

Order 8: $Q_8$, $C_8$

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $Q_{128}$ ($C_1$)

Order 64: $Q_{64}$ ($C_2$) x 2, $C_{64}$ ($C_2$)

Order 32: $C_{32}$ ($C_2^2$)

Order 16: $C_{16}$ ($D_4$)

Order 8: $C_8$ ($D_8$)

Order 4: $C_4$ ($D_{16}$)

Order 2: $C_2$ ($D_{32}$)

Order 1: $C_1$ ($Q_{128}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $Q_{128}$ ($C_1$)

Order 64: $Q_{64}$ ($C_2$), $C_{64}$ ($C_2$)

Order 32: $C_{32}$ ($C_2^2$)

Order 16: $C_{16}$ ($D_4$)

Order 8: $C_8$ ($D_8$)

Order 4: $C_4$ ($D_{16}$)

Order 2: $C_2$ ($D_{32}$)

Order 1: $C_1$ ($Q_{128}$)

Series

Derived series

$Q_{128}$

$\rhd$

$C_{32}$

$\rhd$

$C_1$

Chief series

$Q_{128}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$Q_{128}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

$\lhd$

$C_{16}$

$\lhd$

$C_{32}$

$\lhd$

$Q_{128}$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $35 \times 35$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	4B	4C	8A	16A	32A	64A
	Size	1	1	2	32	32	4	8	16	32
	2 P	1A	1A	2A	2A	2A	4A	8A	16A	32A
	Schur
128.163.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.163.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
128.163.1c		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$
128.163.1d		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$
128.163.2a		$2$	$2$	$2$	$0$	$0$	$2$	$2$	$-2$	$0$
128.163.2b		$4$	$4$	$4$	$0$	$0$	$4$	$-4$	$0$	$0$
128.163.2c		$8$	$8$	$8$	$0$	$0$	$-8$	$0$	$0$	$0$
128.163.2d		$16$	$16$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$
128.163.2e	2	$32$	$-32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$