Abstract group 928.1: $C_{29}:C_{32}$

• Label: 928.1
• Order: \( 2^{5} \cdot 29 \)
• Exponent: \( 2^{5} \cdot 29 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 7 \cdot 29 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 7 \)
• Perm deg.: $61$
• Trans deg.: $928$
• Rank: $2$

Group information

Description:	$C_{29}:C_{32}$
Order:	\(928\)\(\medspace = 2^{5} \cdot 29 \)
Exponent:	\(928\)\(\medspace = 2^{5} \cdot 29 \)
Automorphism group:	$C_{29}.(C_4\times C_{56}).C_2$, of order \(12992\)\(\medspace = 2^{6} \cdot 7 \cdot 29 \)
Composition factors:	$C_2$ x 5, $C_{29}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	4	8	16	29	32	58	116	232	464
Elements	1	1	2	4	8	28	464	28	56	112	224	928
Conjugacy classes	1	1	2	4	8	14	16	14	28	56	112	256
Divisions	1	1	1	1	1	1	1	1	1	1	1	11
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	11

Dimension	1	2	4	8	16	28	56	112	224
Irr. complex chars.	32	224	0	0	0	0	0	0	0	256
Irr. rational chars.	2	1	1	1	1	2	1	1	1	11

Minimal presentations

Permutation degree:	$61$
Transitive degree:	$928$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	448
Arbitrary	2	4	44

Constructions

Presentation:	$\langle a, b \mid a^{32}=b^{29}=1, b^{a}=b^{28} \rangle$
Permutation group:	Degree $61$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 577 & 0 \\ 0 & 256 \end{array}\right), \left(\begin{array}{rr} 0 & 889 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 522 & 0 \\ 0 & 149 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{929})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{29}$ $\,\rtimes\,$ $C_{32}$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{16}$ . $D_{29}$	$C_{464}$ . $C_2$	$C_{232}$ . $C_4$	$C_{116}$ . $C_8$	all 8

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

Homology

Abelianization:	$C_{32} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 40 subgroups in 12 conjugacy classes, 11 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{16}$	$G/Z \simeq$ $D_{29}$
Commutator:	$G' \simeq$ $C_{29}$	$G/G' \simeq$ $C_{32}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $D_{29}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{464}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{29}:C_{32}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{58}$	$G/\operatorname{soc} \simeq$ $C_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{32}$
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 928: $C_{29}:C_{32}$

Order 464: $C_{464}$

Order 232: $C_{232}$

Order 116: $C_{116}$

Order 58: $C_{58}$

Order 32: $C_{32}$

Order 29: $C_{29}$

Order 16: $C_{16}$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 928: $C_{29}:C_{32}$

Order 464: $C_{464}$

Order 232: $C_{232}$

Order 116: $C_{116}$

Order 58: $C_{58}$

Order 32: $C_{32}$

Order 29: $C_{29}$

Order 16: $C_{16}$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 928: $C_{29}:C_{32}$ ($C_1$)

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

Order 232: $C_{232}$ ($C_4$)

Order 116: $C_{116}$ ($C_8$)

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

Order 29: $C_{29}$ ($C_{32}$)

Order 16: $C_{16}$ ($D_{29}$)

Order 8: $C_8$ ($C_{29}:C_4$)

Order 4: $C_4$ ($C_{29}:C_8$)

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

Order 1: $C_1$ ($C_{29}:C_{32}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 928: $C_{29}:C_{32}$ ($C_1$)

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

Order 232: $C_{232}$ ($C_4$)

Order 116: $C_{116}$ ($C_8$)

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

Order 29: $C_{29}$ ($C_{32}$)

Order 16: $C_{16}$ ($D_{29}$)

Order 8: $C_8$ ($C_{29}:C_4$)

Order 4: $C_4$ ($C_{29}:C_8$)

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

Order 1: $C_1$ ($C_{29}:C_{32}$)

Series

Derived series

$C_{29}:C_{32}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Chief series

$C_{29}:C_{32}$

$\rhd$

$C_{464}$

$\rhd$

$C_{232}$

$\rhd$

$C_{116}$

$\rhd$

$C_{58}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

Lower central series

$C_{29}:C_{32}$

$\rhd$

$C_{29}$

Upper central series

$C_1$

$\lhd$

$C_{16}$

Supergroups

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

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $256 \times 256$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	8A	16A	29A	32A	58A	116A	232A	464A
	Size	1	1	2	4	8	28	464	28	56	112	224
	2 P	1A	1A	2A	4A	8A	29A	16A	29A	58A	116A	232A
	29 P	1A	2A	4A	8A	16A	29A	32A	58A	116A	232A	464A
	Schur
928.1.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
928.1.1b		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$
928.1.1c		$2$	$2$	$2$	$2$	$-2$	$2$	$0$	$2$	$2$	$2$	$-2$
928.1.1d		$4$	$4$	$4$	$-4$	$0$	$4$	$0$	$4$	$4$	$-4$	$0$
928.1.1e		$8$	$8$	$-8$	$0$	$0$	$8$	$0$	$8$	$-8$	$0$	$0$
928.1.1f		$16$	$-16$	$0$	$0$	$0$	$16$	$0$	$-16$	$0$	$0$	$0$
928.1.2a		$28$	$28$	$28$	$28$	$28$	$-1$	$0$	$-1$	$-1$	$-1$	$-1$
928.1.2b	2	$28$	$28$	$28$	$28$	$-28$	$-1$	$0$	$-1$	$-1$	$-1$	$1$
928.1.2c	2	$56$	$56$	$56$	$-56$	$0$	$-2$	$0$	$-2$	$-2$	$2$	$0$
928.1.2d	2	$112$	$112$	$-112$	$0$	$0$	$-4$	$0$	$-4$	$4$	$0$	$0$
928.1.2e	2	$224$	$-224$	$0$	$0$	$0$	$-8$	$0$	$8$	$0$	$0$	$0$