C₇×#7229 #13184445

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7229

Subgroups

	subgroup	G/H
center	C₄×C₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₇
Frattini, cosocle	C₂×#131	C₂²×C₇
socle	C₂×C₇	#1289
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ G

244 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,12,17,14,19,26,42)(10,20,22,44,24,46,47,63)(11,25,27,16,30,39,29,41)(13,32,31,43,33,61,55,18)(15,34,35,21,38,48,37,50)(23,52,51,67,53,71,68,45)(28,57,56,54,59,40,58,66)(36,49,64,69,65,70,60,62);(8,10,13,23,26,47,55,68,14,24,33,53,12,22,31,51)(9,15,18,36,17,37,32,60,19,38,43,65,42,35,61,64)(11,21,28,49,27,48,56,69,30,50,59,70,29,34,58,62)(16,20,40,45,25,44,54,52,41,46,57,67,39,63,66,71)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 217);