C₇×#12883 #13190099

order	1792
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#12883

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

342 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,20,25,21,24,29,31,22,26,28,23,19,30,18,27);(9,12)(13,15)(16,18,19,28,22,29,21,20)(17,23,24,27,26,25,30,31);(8,10,11,14)(9,13,12,15)(16,18,19,28,22,29,21,20)(17,23,24,27,26,25,30,31)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5871);