C₇×#7072 #13184288

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7072

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#102
commutator	C₂×C₈	C₂×C₇×C₈
Frattini, cosocle	C₈²	C₂²×C₇
socle	C₂²×C₇	D₄⋊C₈
prime radical	C₂×C₇×D₈	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₇×C₈² ◁ G

146 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 39
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 2 generators on 39 points:

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

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 60);