C₇×#13646 #13190862

order	1792
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13646

Subgroups

	subgroup	G/H
center	C₂×C₇×C₁₆	C₂³
commutator	C₂²	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₂²×C₁₆
prime radical	C₂×C₇×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₁₆ ◁ G

202 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 43
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,16,21,17,22,30,37,18,14,23,25,31,26,38,40,15,19,27,32,28,33,41,42,29,24,20,34,35,39,36,43);(9,11)(13,19)(14,24)(21,32)(22,33)(25,34)(26,39)(37,42)(40,43);(12,14,16,25,17,26,30,40,18,19,23,32,31,33,38,42,15,24,27,34,28,39,41,43,29,13,20,21,35,22,36,37)

Conjugacy classes & character table

784 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6634);