Abstract group 332.4: $C_2\times C_{166}$

• Label: 332.4
• Order: \( 2^{2} \cdot 83 \)
• Exponent: \( 2 \cdot 83 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3 \cdot 41 \)
• Perm deg.: $87$
• Trans deg.: $332$
• Rank: $2$

Group information

Description:	$C_2\times C_{166}$
Order:	\(332\)\(\medspace = 2^{2} \cdot 83 \)
Exponent:	\(166\)\(\medspace = 2 \cdot 83 \)
Automorphism group:	$S_3\times C_{82}$, of order \(492\)\(\medspace = 2^{2} \cdot 3 \cdot 41 \)
Composition factors:	$C_2$ x 2, $C_{83}$
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	83	166
Elements	1	3	82	246	332
Conjugacy classes	1	3	82	246	332
Divisions	1	3	1	3	8
Autjugacy classes	1	1	1	1	4

Dimension	1	82
Irr. complex chars.	332	0	332
Irr. rational chars.	4	4	8

Minimal presentations

Permutation degree:	$87$
Transitive degree:	$332$
Rank:	$2$
Inequivalent generating pairs:	$84$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	83

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{166}=1 \rangle$
Permutation group:	Degree $87$ $\langle(1,2), (3,4), (5,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 52 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 166 & 0 \\ 0 & 166 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{167})$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_{83}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_{501})$	$\Aut(C_{668})$	$\Aut(C_{1002})$

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

Homology

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

Subgroups

There are 10 subgroups, all normal (4 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{166}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{166}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times C_{166}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{166}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{166}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{166}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
83-Sylow subgroup:	$P_{ 83 } \simeq$ $C_{83}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 332: $C_2\times C_{166}$

Order 166: $C_{166}$ x 3

Order 83: $C_{83}$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 332: $C_2\times C_{166}$

Order 166: $C_{166}$

Order 83: $C_{83}$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 332: $C_2\times C_{166}$ ($C_1$)

Order 166: $C_{166}$ ($C_2$) x 3

Order 83: $C_{83}$ ($C_2^2$)

Order 4: $C_2^2$ ($C_{83}$)

Order 2: $C_2$ ($C_{166}$) x 3

Order 1: $C_1$ ($C_2\times C_{166}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 332: $C_2\times C_{166}$ ($C_1$)

Order 166: $C_{166}$ ($C_2$)

Order 83: $C_{83}$ ($C_2^2$)

Order 4: $C_2^2$ ($C_{83}$)

Order 2: $C_2$ ($C_{166}$)

Order 1: $C_1$ ($C_2\times C_{166}$)

Series

Derived series	$C_2\times C_{166}$	$\rhd$	$C_1$
Chief series	$C_2\times C_{166}$	$\rhd$	$C_{166}$	$\rhd$	$C_{83}$	$\rhd$	$C_1$
Lower central series	$C_2\times C_{166}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2\times C_{166}$

Supergroups

This group is a maximal subgroup of 18 larger groups in the database.

This group is a maximal quotient of 15 larger groups in the database.

Character theory

Complex character table

See the $332 \times 332$ 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	2B	2C	83A	166A	166B	166C
	Size	1	1	1	1	82	82	82	82
	2 P	1A	1A	1A	1A	83A	83A	83A	83A
	83 P	1A	2A	2B	2C	1A	2A	2B	2C
332.4.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
332.4.1b		$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
332.4.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
332.4.1d		$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$
332.4.1e		$82$	$82$	$82$	$82$	$-1$	$-1$	$-1$	$-1$
332.4.1f		$82$	$-82$	$-82$	$82$	$-1$	$-1$	$-1$	$-1$
332.4.1g		$82$	$-82$	$82$	$-82$	$-1$	$1$	$1$	$1$
332.4.1h		$82$	$82$	$-82$	$-82$	$-1$	$1$	$1$	$1$