C₇×C₈×Q₃₂ #13189258

order	1792
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×C₈×Q₃₂

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	D₈
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle, prime radical	C₂²×C₇	C₄×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈ ◁ C₇×C₈² ◁ G

226 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 47
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15);(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33);(16,32,24,40)(17,33,25,41)(18,34,26,42)(19,35,27,43)(20,36,28,44)(21,37,29,45)(22,38,30,46)(23,39,31,47)

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, 5030);