C₇×#12046 #13189262

order	1792
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12046

Subgroups

	subgroup	G/H
center	C₇×C₁₆	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×C₁₆ ◁ C₂×C₇×C₁₆ ◁ C₄×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 216 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)(8,9,13,19,15,20,31,18,16,21,12,30,14,10,22,24)(11,17,26,33,28,34,39,32,29,35,25,38,27,23,36,37);(8,10)(9,15)(11,23)(12,18)(13,24)(14,21)(16,20)(17,28)(19,31)(22,30)(25,32)(26,37)(27,35)(29,34)(33,39)(36,38);(8,11,13,26,15,28,31,39,16,29,12,25,14,27,22,36)(9,17,19,33,20,34,18,32,21,35,30,38,10,23,24,37)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5034);