C₇×#12730 #13189946

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#12730

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1782
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×Q₈	C₂³×C₇
prime radical	C₂×C₇×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₇×#519 ◁ G

666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9,13,18,14,19,10,20)(11,23,26,34,25,16,21,32)(12,17,22,30,28,33,27,15)(24,37,35,29,39,31,38,36);(8,10,14,13)(9,15,19,30)(11,21,25,26)(12,27,28,22)(16,29,23,36)(17,18,33,20)(24,38,39,35)(31,32,37,34);(8,11,14,25)(9,16)(10,21,13,26)(12,24,28,39)(15,29)(17,31)(18,32)(19,23)(20,34)(22,35,27,38)(30,36)(33,37)

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, 5718);