C₇×#13577 #13190793

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#13577

Subgroups

	subgroup	G/H
center	C₄×C₇	#353
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
socle	C₂×C₇	C₂×#353
prime radical	C₇×#3150	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)(10,20,22,36)(11,16)(12,26)(13,18,29,33)(14,19)(15,21,31,38)(17,28,34,25)(23,30,24,37)(27,35,39,32);(8,10)(9,15)(11,21)(12,27)(13,22)(14,24)(16,30)(17,20)(18,35)(19,32)(23,29)(25,38)(26,36)(28,39)(31,33)(34,37);(8,11,14,12)(9,16,19,26)(10,21,24,27)(13,25,29,28)(15,30,32,36)(17,33,34,18)(20,31,37,35)(22,38,23,39)

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, 6565);