C₇×#11859 #13189075

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#11859

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

314 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 448

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

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

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, 4847);