C₇×#12845 #13190061

order	1792
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12845

Subgroups

	subgroup	G/H
center	C₂²×C₇×C₈	C₂³
commutator	C₂³	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ G

454 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 51
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19)(20,21,23,27,24,28,33,39,25,29,34,40,35,41,45,49)(22,26,31,37,32,38,44,48,30,36,42,46,43,47,50,51);(13,16)(17,19)(20,22)(21,26)(23,31)(24,32)(25,30)(27,37)(28,38)(29,36)(33,44)(34,42)(35,43)(39,48)(40,46)(41,47)(45,50)(49,51);(9,11)(12,14,15,18)(13,17,16,19)(22,30)(26,36)(31,42)(32,43)(37,46)(38,47)(44,50)(48,51)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5833);