C₇×#12862 #13190078

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12862

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁴×C₇ ◁ C₂³×C₇×Q₈ ◁ G

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

normal subgroups only

This group has 566 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) = 47
transitive degree	μ_t(G) = 448

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

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5850);