C₇×#20340 #13197556

order	1792
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₇×#20340

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	C₂²×#95
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ G

1956 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 1114 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 — 4 generators on 39 points:

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

Conjugacy classes & character table

574 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 13328);