C₇×#12287 #13189503

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

Aliases

direct product

C₇×#12287

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₇×#3203	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#408 ◁ G

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

normal subgroups only

This group has 318 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)(26,38)(27,41)(28,44)(29,36)(30,35)(31,37)(32,39)(33,43)(34,47)(40,52)(42,51)(45,54)(46,49)(48,53)(50,55);(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,23)(21,22)(24,26,31,27)(25,32,37,51)(28,45,48,55)(29,40,30,43)(33,49,52,34)(35,50,36,54)(38,53,41,44)(39,46,42,47);(8,11)(9,14)(10,16)(12,18)(13,19)(15,21)(17,23)(20,22)(24,27,31,26)(25,33,37,52)(28,42,48,39)(29,40,30,43)(32,49,51,34)(35,38,36,41)(44,50,53,54)(45,47,55,46)

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, 5275);