C₇×#8799 #13186015

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#8799

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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)(8,9,13,17,14,19,12,18)(10,16,20,23,21,22,11,15)(24,25,28,33,29,32,35,45)(26,34,38,47,39,30,37,48)(27,40,43,31,44,51,42,50)(36,52,53,46,41,55,54,49);(8,10,12,11,14,21,13,20)(9,15,17,16,19,23,18,22)(24,26,35,38,29,39,28,37)(25,30,45,48,32,34,33,47)(27,41,43,53,44,36,42,54)(31,49,51,55,50,46,40,52);(8,11,12,21,14,20,13,10)(9,16,17,23,19,22,18,15)(24,27)(25,31,32,50)(26,36)(28,42)(29,44)(30,46,34,49)(33,40,45,51)(35,43)(37,53)(38,54)(39,41)(47,55,48,52)

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, 1787);