C₇×#8798 #13186014

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×#8798

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

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, 1786);