C₇×#12047 #13189263

order	1792
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12047

Subgroups

	subgroup	G/H
center	C₇×C₈	C₂×D₈
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂×C₇	C₈×D₈
prime radical	C₇×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₁₆ ◁ C₄×C₇×C₁₆ ◁ G

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,9,13,19,15,20,32,18,16,21,12,31,14,10,22,24)(11,25,27,39,29,23,37,38,30,17,26,34,28,35,36,33);(8,10)(9,15)(11,23)(12,18)(13,24)(14,21)(16,20)(17,29)(19,32)(22,31)(25,28)(26,33)(27,38)(30,35)(34,37)(36,39);(8,11,13,27,15,29,32,37,16,30,12,26,14,28,22,36)(9,17,19,34,20,35,18,33,21,25,31,39,10,23,24,38)

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, 5035);