C₇×#8802 #13186018

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#8802

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#383
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

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

normal subgroups only

This group has 258 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) = 39
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9)(10,18,21,22)(11,14)(12,16)(13,17)(15,20,23,19)(24,25,27,30,28,31,34,37)(26,29,33,36,32,35,38,39);(9,14,17,16)(15,22,23,18)(24,26)(25,29)(27,33)(28,32)(30,36)(31,35)(34,38)(37,39);(8,10,11,19,13,21,12,20)(9,15,16,18,17,23,14,22)(26,32)(29,35)(33,38)(36,39)

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, 1790);