C₇×#7468 #13184684

order	1792
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7468

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂²⋊C₄
commutator	C₂²	C₄×C₇×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂²×C₇	C₂²⋊C₁₆
prime radical	C₂³×C₇	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 182 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) = 43
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 2 generators on 43 points:

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

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, 456);