Abstract group 27722.d: $C_{27722}$

• Label: 27722.d
• Order: \( 2 \cdot 83 \cdot 167 \)
• Exponent: \( 2 \cdot 83 \cdot 167 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{2} \cdot 41 \cdot 83 \)
• Perm deg.: $252$
• Trans deg.: $27722$
• Rank: $1$

Group information

Description:	$C_{27722}$
Order:	\(27722\)\(\medspace = 2 \cdot 83 \cdot 167 \)
Exponent:	\(27722\)\(\medspace = 2 \cdot 83 \cdot 167 \)
Automorphism group:	$C_2\times C_{6806}$, of order \(13612\)\(\medspace = 2^{2} \cdot 41 \cdot 83 \)
Composition factors:	$C_2$, $C_{83}$, $C_{167}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,83,167$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	83	166	167	334	13861	27722
Elements	1	1	82	82	166	166	13612	13612	27722
Conjugacy classes	1	1	82	82	166	166	13612	13612	27722
Divisions	1	1	1	1	1	1	1	1	8
Autjugacy classes	1	1	1	1	1	1	1	1	8

Minimal presentations

Permutation degree:	$252$
Transitive degree:	$27722$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	not computed	not computed
Arbitrary	1	not computed	not computed

Constructions

Presentation:	$\langle a \mid a^{27722}=1 \rangle$
Permutation group:	Degree $252$ $\langle(1,2), (3,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 166 & 0 \\ 0 & 166 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{167})$
Direct product:	$C_2$ $\, \times\, $ $C_{83}$ $\, \times\, $ $C_{167}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
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

Primary decomposition:	$C_{2} \times C_{83} \times C_{167}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 8 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{27722}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{27722}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{27722}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{27722}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{27722}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{27722}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
83-Sylow subgroup:	$P_{ 83 } \simeq$ $C_{83}$
167-Sylow subgroup:	$P_{ 167 } \simeq$ $C_{167}$

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 27722: $C_{27722}$

Order 13861: $C_{13861}$

Order 334: $C_{334}$

Order 167: $C_{167}$

Order 166: $C_{166}$

Order 83: $C_{83}$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 27722: $C_{27722}$

Order 13861: $C_{13861}$

Order 334: $C_{334}$

Order 167: $C_{167}$

Order 166: $C_{166}$

Order 83: $C_{83}$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 27722: $C_{27722}$ ($C_1$)

Order 13861: $C_{13861}$ ($C_2$)

Order 334: $C_{334}$ ($C_{83}$)

Order 167: $C_{167}$ ($C_{166}$)

Order 166: $C_{166}$ ($C_{167}$)

Order 83: $C_{83}$ ($C_{334}$)

Order 2: $C_2$ ($C_{13861}$)

Order 1: $C_1$ ($C_{27722}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 27722: $C_{27722}$ ($C_1$)

Order 13861: $C_{13861}$ ($C_2$)

Order 334: $C_{334}$ ($C_{83}$)

Order 167: $C_{167}$ ($C_{166}$)

Order 166: $C_{166}$ ($C_{167}$)

Order 83: $C_{83}$ ($C_{334}$)

Order 2: $C_2$ ($C_{13861}$)

Order 1: $C_1$ ($C_{27722}$)

Series

Derived series	$C_{27722}$	$\rhd$	$C_1$
Chief series	$C_{27722}$	$\rhd$	$C_{13861}$	$\rhd$	$C_{167}$	$\rhd$	$C_1$
Lower central series	$C_{27722}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{27722}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $27722 \times 27722$ character table is not available for this group.

Rational character table

The $8 \times 8$ rational character table is not available for this group.