Abstract group 108811.1: $C_{467}:C_{233}$

• Label: 108811.1
• Order: \( 233 \cdot 467 \)
• Exponent: \( 233 \cdot 467 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 233 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2 \cdot 233 \cdot 467 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $467$
• Trans deg.: $467$
• Rank: $2$

Group information

Description:	$C_{467}:C_{233}$
Order:	\(108811\)\(\medspace = 233 \cdot 467 \)
Exponent:	\(108811\)\(\medspace = 233 \cdot 467 \)
Automorphism group:	$F_{467}$, of order \(217622\)\(\medspace = 2 \cdot 233 \cdot 467 \)
Composition factors:	$C_{233}$, $C_{467}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 233$.

Group statistics

Order	1	233	467
Elements	1	108344	466	108811
Conjugacy classes	1	232	2	235
Divisions	1	1	1	3
Autjugacy classes	1	232	1	234

Dimension	1	232	233	466
Irr. complex chars.	233	0	2	0	235
Irr. rational chars.	1	1	0	1	3

Minimal presentations

Permutation degree:	$467$
Transitive degree:	$467$
Rank:	$2$
Inequivalent generating pairs:	$54288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	233	466	466
Arbitrary	233	466	466

Constructions

Presentation:	$\langle a, b \mid a^{233}=b^{467}=1, b^{a}=b^{177} \rangle$
Permutation group:	Degree $467$ $\langle(2,5,17,65,257,91,361,40,157,158,162,178,242,31,121,14,53,209,366,60,237,11,41,161,174,226,434,332,391,160,170,210,370,76,301,267,131,54,213,382,124,26,101,401,200,330,383,128,42,165,190,290,223,422,284,199,326,367,64,253,75,297,251,67,265,123,22,85,337,411,240,23,89,353,8,29,113,449,392,164,186,274,159,166,194,306,287,211,374,92,365,56,221,414,252,71,281,187,278,175,230,450,396,180,250,63,249,59,233,462,444,372,84,333,395,176,234,466,460,436,340,423,288,215,390,156,154,146,114,453,408,228,442,364,52,205,350,463,448,388,148,122,18,69,273,155,150,130,50,197,318,335,403,208,362,44,173,222,418,268,135,70,277,171,214,386,140,90,357,24,93,369,72,285,203,342,431,320,343,435,336,407,224,426,300,263,115,457,424,292,231,454,412,244,39,153,142,98,389,152,138,82,325,363,48,189,286,207,358,28,109,433,328,375,96,381,120,10,37,145,110,437,344,439,352,4,13,49,193,302,271,147,118) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 4 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{467})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{467}$ $\,\rtimes\,$ $C_{233}$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{233} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 470 subgroups in 4 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{467}:C_{233}$
Commutator:	$G' \simeq$ $C_{467}$	$G/G' \simeq$ $C_{233}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{467}:C_{233}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{467}$	$G/\operatorname{Fit} \simeq$ $C_{233}$
Radical:	$R \simeq$ $C_{467}:C_{233}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{467}$	$G/\operatorname{soc} \simeq$ $C_{233}$
233-Sylow subgroup:	$P_{ 233 } \simeq$ $C_{233}$
467-Sylow subgroup:	$P_{ 467 } \simeq$ $C_{467}$

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 108811: $C_{467}:C_{233}$

Order 467: $C_{467}$

Order 233: $C_{233}$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 108811: $C_{467}:C_{233}$

Order 467: $C_{467}$

Order 233: $C_{233}$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 108811: $C_{467}:C_{233}$ ($C_1$)

Order 467: $C_{467}$ ($C_{233}$)

Order 1: $C_1$ ($C_{467}:C_{233}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 108811: $C_{467}:C_{233}$ ($C_1$)

Order 467: $C_{467}$ ($C_{233}$)

Order 1: $C_1$ ($C_{467}:C_{233}$)

Series

Derived series	$C_{467}:C_{233}$	$\rhd$	$C_{467}$	$\rhd$	$C_1$
Chief series	$C_{467}:C_{233}$	$\rhd$	$C_{467}$	$\rhd$	$C_1$
Lower central series	$C_{467}:C_{233}$	$\rhd$	$C_{467}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $235 \times 235$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	233A	467A
	Size	1	108344	466
	233 P	1A	233A	467A
	467 P	1A	233A	467A
108811.1.1a		$1$	$1$	$1$
108811.1.1b		$232$	$-1$	$232$
108811.1.233a		$466$	$0$	$-1$