C₇×#13658 #13190874

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13658

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2177
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₇×#2263	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₇×#346 ◁ G

194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,15,17,31)(10,20)(11,23)(12,27)(13,21)(16,24,28,32)(18,30,35,34)(19,36,29,33)(22,26)(25,37)(38,39);(8,9)(10,16)(11,24)(12,28)(13,29)(14,17)(15,22)(18,27)(19,21)(20,35)(23,31)(25,30)(26,32)(33,39)(34,38)(36,37);(8,10,21,27,14,12,13,20)(9,16,19,18,17,28,29,35)(11,23,38,37,26,22,25,39)(15,30,33,24,31,34,36,32)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6646);