C₇×#13578 #13190794

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13578
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13578

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×#353
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₇×#3153	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#124 ◁ G

420 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 90 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)(10,18,21,33)(11,22,23,38)(12,25,27,16)(13,17)(14,20,29,36)(15,24)(19,31,26,34)(28,35,30,37)(32,39);(8,10)(9,14)(11,20)(12,26)(13,19)(15,31)(16,29)(17,28)(18,32)(21,27)(22,34)(23,35)(24,36)(25,30)(33,38)(37,39);(10,19)(14,28)(18,31)(20,35)(21,26)(29,30)(33,34)(36,37)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6566);