C₇×#8540 #13185756

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#8540

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₇×#534	C₄
LCS	G ▷ C₂⁴ ▷ 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)(24,25,28,32,29,33,39,45)(26,30,36,42,35,41,49,53)(27,31,38,44,37,43,50,54)(34,46,48,55,47,40,51,52);(8,10)(9,13)(11,17)(12,18)(14,20)(15,16)(19,23)(21,22)(24,26)(25,30)(27,34)(28,36)(29,35)(31,40)(32,42)(33,41)(37,47)(38,48)(39,49)(43,46)(44,52)(45,53)(50,51)(54,55);(8,11,12,19)(9,14,15,21)(10,17,18,23)(13,20,16,22)(24,27)(25,31)(26,34)(28,38)(29,37)(30,40)(32,44)(33,43)(35,47)(36,48)(39,50)(41,46)(42,52)(45,54)(49,51)(53,55)

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, 1528);