C₇×#13589 #13190805

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13589

Subgroups

	subgroup	G/H
center	C₄×C₇	#353
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
socle	C₂×C₇	C₂×#353
prime radical	C₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

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

normal subgroups only

This group has 408 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)(8,9,12,17,13,18,28,35)(10,19,21,34,23,30,37,38)(11,15,24,32,25,16,27,33)(14,26,29,20,31,36,39,22);(8,10)(9,14)(11,20)(12,26)(13,23)(15,19)(16,30)(17,34)(18,31)(21,24)(22,25)(27,37)(28,36)(29,33)(32,39)(35,38);(8,11,13,25)(9,15,18,16)(10,20,23,22)(12,24,28,27)(14,19,31,30)(17,32,35,33)(21,36,37,26)(29,34,39,38)

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, 6577);