C₇×#8463 #13185679

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#8463

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₇×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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)(17,22,23,20)(19,21)(24,25,29,34,30,35,28,33)(26,36,40,31,39,46,38,45)(27,32,41,47,42,48,43,49)(37,51,52,55,53,50,54,44);(8,10)(9,13)(11,17)(12,18)(14,20)(15,16)(19,23)(21,22)(24,26)(25,31)(27,37)(28,40)(29,38)(30,39)(32,44)(33,46)(34,36)(35,45)(41,54)(42,53)(43,52)(47,50)(48,55)(49,51);(8,11,12,19)(9,14,15,21)(10,17,18,23)(13,20,16,22)(24,27)(25,32)(26,37)(28,43)(29,41)(30,42)(31,44)(33,49)(34,47)(35,48)(36,50)(38,54)(39,53)(40,52)(45,55)(46,51)

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, 1451);