C₇×#20773 #13197989

order	1792
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#20773

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

830 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 382 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) = 896

A minimal faithful permutation representation — 4 generators on 39 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 13761);