C₇×#11967 #13189183

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11967

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

290 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 122 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)(24,25,28,33,29,34,43,51,30,35,44,52,42,50,27,32)(26,36,38,48,39,49,37,53,40,54,41,31,45,46,55,47);(8,10,12,18)(9,13,14,16)(11,19,20,17)(15,23,22,21)(24,26)(25,31)(27,41)(28,37)(29,45)(30,40)(32,49)(33,36)(34,53)(35,48)(38,43)(39,42)(44,55)(46,51)(47,50)(52,54);(9,14)(13,16)(15,22)(21,23)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4955);