C₇×#8477 #13185693

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#8477

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 682 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)(8,9,11,15,12,14,20,22)(10,16,17,23,18,13,19,21)(24,25)(26,34)(27,38,40,31)(28,41)(29,33)(30,36)(32,44)(35,50,52,55)(37,48)(39,47,43,49)(42,46)(45,51,54,53);(8,10,12,18)(9,13,14,16)(11,19,20,17)(15,23,22,21)(24,26,29,37)(25,30,33,46)(27,35,40,52)(28,42,44,36)(31,45,38,54)(32,48,41,34)(39,53,43,51)(47,50,49,55);(9,14)(11,20)(13,16)(17,19)(24,27)(25,31)(26,35)(28,43)(29,40)(30,45)(32,49)(33,38)(34,50)(36,53)(37,52)(39,44)(41,47)(42,51)(46,54)(48,55)

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, 1465);