Abstract group 1856.54: $C_{29}\times Q_{64}$

• Label: 1856.54
• Order: \( 2^{6} \cdot 29 \)
• Exponent: \( 2^{5} \cdot 29 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 29 \)
• $\card{Z(G)}$: \( 2 \cdot 29 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 7 \)
• Perm deg.: $93$
• Trans deg.: $1856$
• Rank: $2$

Group information

Description:	$C_{29}\times Q_{64}$
Order:	\(1856\)\(\medspace = 2^{6} \cdot 29 \)
Exponent:	\(928\)\(\medspace = 2^{5} \cdot 29 \)
Automorphism group:	$C_{28}\times C_8.(C_8\times D_4)$, of order \(14336\)\(\medspace = 2^{11} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_{29}$
Nilpotency class:	$5$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).

Group statistics

Order	1	2	4	8	16	29	32	58	116	232	464	928
Elements	1	1	34	4	8	28	16	28	952	112	224	448	1856
Conjugacy classes	1	1	3	2	4	28	8	28	84	56	112	224	551
Divisions	1	1	3	1	1	1	1	1	3	1	1	1	16
Autjugacy classes	1	1	2	1	1	1	1	1	2	1	1	1	14

Dimension	1	2	4	8	16	28	56	112	224	448
Irr. complex chars.	116	435	0	0	0	0	0	0	0	0	551
Irr. rational chars.	4	1	1	1	1	4	1	1	1	1	16

Minimal presentations

Permutation degree:	$93$
Transitive degree:	$1856$
Rank:	$2$
Inequivalent generating pairs:	$90$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	448
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{928}=1, a^{2}=b^{464}, b^{a}=b^{639} \rangle$
Permutation group:	Degree $93$ $\langle(1,34,2,33)(3,36,4,35)(5,38,6,37)(7,40,8,39)(9,42,10,41)(11,44,12,43)(13,46,14,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 577 & 0 \\ 0 & 577 \end{array}\right), \left(\begin{array}{rr} 701 & 0 \\ 0 & 656 \end{array}\right), \left(\begin{array}{rr} 0 & 577 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{929})$
Direct product:	$C_{29}$ $\, \times\, $ $Q_{64}$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_{32}$ . $C_{58}$ (2)	$C_{232}$ . $D_4$	$C_{116}$ . $D_8$	$C_{58}$ . $D_{16}$	all 12

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

Homology

Abelianization:	$C_{2} \times C_{58} \simeq C_{2}^{2} \times C_{29}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 74 subgroups in 30 conjugacy classes, 18 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{58}$	$G/Z \simeq$ $D_{16}$
Commutator:	$G' \simeq$ $C_{16}$	$G/G' \simeq$ $C_2\times C_{58}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_2\times C_{58}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{29}\times Q_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{29}\times Q_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{58}$	$G/\operatorname{soc} \simeq$ $D_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{64}$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$

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 1856: $C_{29}\times Q_{64}$

Order 928: $C_{29}\times Q_{32}$ x 2, $C_{928}$

Order 464: $Q_{16}\times C_{29}$ x 2, $C_{464}$

Order 232: $Q_8\times C_{29}$ x 2, $C_{232}$

Order 116: $C_{116}$ x 3

Order 64: $Q_{64}$

Order 58: $C_{58}$

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

Order 29: $C_{29}$

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 1856: $C_{29}\times Q_{64}$

Order 928: $C_{29}\times Q_{32}$, $C_{928}$

Order 464: $Q_{16}\times C_{29}$, $C_{464}$

Order 232: $C_{232}$, $Q_8\times C_{29}$

Order 116: $C_{116}$ x 2

Order 64: $Q_{64}$

Order 58: $C_{58}$

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

Order 29: $C_{29}$

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 1856: $C_{29}\times Q_{64}$ ($C_1$)

Order 928: $C_{29}\times Q_{32}$ ($C_2$) x 2, $C_{928}$ ($C_2$)

Order 464: $C_{464}$ ($C_2^2$)

Order 232: $C_{232}$ ($D_4$)

Order 116: $C_{116}$ ($D_8$)

Order 64: $Q_{64}$ ($C_{29}$)

Order 58: $C_{58}$ ($D_{16}$)

Order 32: $Q_{32}$ ($C_{58}$) x 2, $C_{32}$ ($C_{58}$)

Order 29: $C_{29}$ ($Q_{64}$)

Order 16: $C_{16}$ ($C_2\times C_{58}$)

Order 8: $C_8$ ($D_4\times C_{29}$)

Order 4: $C_4$ ($D_8\times C_{29}$)

Order 2: $C_2$ ($C_{29}\times D_{16}$)

Order 1: $C_1$ ($C_{29}\times Q_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1856: $C_{29}\times Q_{64}$ ($C_1$)

Order 928: $C_{29}\times Q_{32}$ ($C_2$), $C_{928}$ ($C_2$)

Order 464: $C_{464}$ ($C_2^2$)

Order 232: $C_{232}$ ($D_4$)

Order 116: $C_{116}$ ($D_8$)

Order 64: $Q_{64}$ ($C_{29}$)

Order 58: $C_{58}$ ($D_{16}$)

Order 32: $Q_{32}$ ($C_{58}$), $C_{32}$ ($C_{58}$)

Order 29: $C_{29}$ ($Q_{64}$)

Order 16: $C_{16}$ ($C_2\times C_{58}$)

Order 8: $C_8$ ($D_4\times C_{29}$)

Order 4: $C_4$ ($D_8\times C_{29}$)

Order 2: $C_2$ ($C_{29}\times D_{16}$)

Order 1: $C_1$ ($C_{29}\times Q_{64}$)

Series

Derived series

$C_{29}\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_1$

Chief series

$C_{29}\times Q_{64}$

$\rhd$

$C_{29}\times Q_{32}$

$\rhd$

$C_{464}$

$\rhd$

$C_{232}$

$\rhd$

$C_{116}$

$\rhd$

$C_{58}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Lower central series

$C_{29}\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{58}$

$\lhd$

$C_{116}$

$\lhd$

$C_{232}$

$\lhd$

$C_{464}$

$\lhd$

$C_{29}\times Q_{64}$

Character theory

Complex character table

The $551 \times 551$ character table is not available for this group.

Rational character table

		1A	2A	4A	4B	4C	8A	16A	29A	32A	58A	116A	116B	116C	232A	464A	928A
	Size	1	1	2	16	16	4	8	28	16	28	56	448	448	112	224	448
	2 P	1A	1A	2A	2A	2A	4A	8A	29A	16A	29A	58A	58A	58A	116A	232A	464A
	29 P	1A	2A	4A	4B	4C	8A	16A	29A	32A	58A	116A	116B	116C	232A	464A	928A
	Schur
1856.54.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1856.54.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
1856.54.1c		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$
1856.54.1d		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
1856.54.1e		$28$	$28$	$28$	$28$	$28$	$28$	$28$	$-1$	$28$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
1856.54.1f		$28$	$28$	$28$	$-28$	$-28$	$28$	$28$	$-1$	$28$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
1856.54.1g		$28$	$28$	$28$	$-28$	$28$	$28$	$28$	$-1$	$-28$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$
1856.54.1h		$28$	$28$	$28$	$28$	$-28$	$28$	$28$	$-1$	$-28$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
1856.54.2a		$2$	$2$	$2$	$0$	$0$	$2$	$-2$	$2$	$0$	$2$	$2$	$0$	$0$	$2$	$-2$	$0$
1856.54.2b		$4$	$4$	$4$	$0$	$0$	$-4$	$0$	$4$	$0$	$4$	$4$	$0$	$0$	$-4$	$0$	$0$
1856.54.2c		$8$	$8$	$-8$	$0$	$0$	$0$	$0$	$8$	$0$	$8$	$-8$	$0$	$0$	$0$	$0$	$0$
1856.54.2d	2	$16$	$-16$	$0$	$0$	$0$	$0$	$0$	$16$	$0$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$
1856.54.2e		$56$	$56$	$56$	$0$	$0$	$56$	$-56$	$-2$	$0$	$-2$	$-2$	$0$	$0$	$-2$	$2$	$0$
1856.54.2f		$112$	$112$	$112$	$0$	$0$	$-112$	$0$	$-4$	$0$	$-4$	$-4$	$0$	$0$	$4$	$0$	$0$
1856.54.2g		$224$	$224$	$-224$	$0$	$0$	$0$	$0$	$-8$	$0$	$-8$	$8$	$0$	$0$	$0$	$0$	$0$
1856.54.2h		$448$	$-448$	$0$	$0$	$0$	$0$	$0$	$-16$	$0$	$16$	$0$	$0$	$0$	$0$	$0$	$0$