C₇×#7467 #13184683

order	1792
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7467

Subgroups

	subgroup	G/H
center	C₂×C₇×C₁₆	D₄
commutator	C₄	C₄×C₇×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂²×C₇	C₄⋊C₁₆
prime radical	C₂³×C₇	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₁₆ ◁ C₂²×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 190 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) = 448

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

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

Conjugacy classes & character table

784 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 455);