C₇×#11970 #13189186

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11970

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 284 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) = 55
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,11,15,12,14,20,22)(10,16,17,23,18,13,19,21)(24,25,29,35,30,33,45,51,31,36,43,53,27,42,28,34)(26,37,40,50,38,52,39,47,41,54,44,32,46,48,55,49);(8,10,12,18)(9,13,14,16)(11,19,20,17)(15,23,22,21)(24,26)(25,32)(27,38)(28,44)(29,39)(30,46)(31,41)(33,47)(34,52)(35,37)(36,50)(40,45)(42,49)(43,55)(48,51)(53,54);(9,14)(13,16)(15,22)(21,23)(24,27,31,30)(25,33,36,42)(26,38,41,46)(28,43,45,29)(32,47,50,49)(34,35,51,53)(37,48,54,52)(39,44,55,40)

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, 4958);