C₇×#13590 #13190806

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13590

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×#353
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₇×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₄⋊Q₈ ◁ G

398 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6578);