C₇×#12294 #13189510

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12294

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₇×#3259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#424 ◁ G

302 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,9,12,15)(10,13,17,20)(11,14,18,21)(16,22,23,19)(24,25,30,36)(26,37,40,52)(27,31,44,48)(28,35,29,34)(32,45,50,39)(33,43,47,41)(38,51,42,49)(46,54,53,55);(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,23)(21,22)(25,31)(27,41)(28,29)(32,46)(33,35)(34,47)(36,48)(37,51)(38,54)(39,45)(42,55)(43,44)(49,52)(50,53);(8,11,12,18)(9,14,15,21)(10,16,17,23)(13,19,20,22)(24,26,30,40)(25,32,36,50)(27,42,44,38)(28,45,29,39)(31,46,48,53)(33,51,47,49)(34,52,35,37)(41,55,43,54)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5282);