C₇×#13616 #13190832

order	1792
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13616

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂×C₇×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 302 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)(13,18)(14,22)(15,24)(16,28)(19,31)(20,34)(21,35)(23,27)(25,41)(26,38)(29,40)(30,32)(33,42)(36,37)(39,43);(12,13)(14,23)(15,27)(16,30)(17,19)(18,22)(20,24)(21,28)(25,42)(26,39)(29,43)(31,38)(32,41)(33,40)(34,36)(35,37);(13,19)(18,31)(20,33)(21,30)(23,39)(27,43)(32,35)(34,42)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6604);