C₇×#9425 #13186641

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9425

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 534 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) = 39
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2413);