C₇×#8797 #13186013

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#8797

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

228 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,9,12,17,13,18,20,16)(10,19,11,14,21,15,22,23)(24,25,28,33,29,32,35,44)(26,34,38,46,39,30,37,47)(27,40,42,31,43,49,41,48)(36,51,52,45,53,55,54,50);(8,10,20,22,13,21,12,11)(9,14,17,15,18,23,16,19)(24,26,35,38,29,39,28,37)(25,30,44,47,32,34,33,46)(27,36,42,54,43,53,41,52)(31,45,49,51,48,50,40,55);(8,11,12,21,13,22,20,10)(9,15,16,14,18,19,17,23)(24,27)(25,31,32,48)(26,36)(28,41)(29,43)(30,45,34,50)(33,40,44,49)(35,42)(37,52)(38,54)(39,53)(46,55,47,51)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1785);