C₇×#11835 #13189051

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11835

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

340 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(9,13)(10,11)(14,15)(16,17,20,25,21,24,33,40)(18,26,28,41,29,22,34,36)(19,23,31,39,32,38,44,47)(27,37,42,46,30,35,43,45);(8,9)(10,13)(11,15)(12,14)(16,18,21,29)(17,22,24,26)(19,30,32,27)(20,28,33,34)(23,37,38,35)(25,41,40,36)(31,43,44,42)(39,45,47,46);(8,10,12,11)(9,13,14,15)(16,19,21,32)(17,23,24,38)(18,27,29,30)(20,31,33,44)(22,35,26,37)(25,39,40,47)(28,42,34,43)(36,45,41,46)

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, 4823);