C₇×#13603 #13190819

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13603

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×#354
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#124 ◁ G

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

normal subgroups only

This group has 334 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)(10,18,21,34)(11,22,24,38)(12,26,28,16)(13,17)(14,20,30,36)(15,25)(19,33,27,35)(23,31,37,29)(32,39);(8,10)(9,14,17,29)(11,23)(12,27)(13,19)(15,18,32,33)(16,30,26,31)(20,24)(21,28)(22,34,38,35)(25,37)(36,39);(10,19)(14,29)(18,33)(20,23)(21,27)(30,31)(34,35)(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, 6591);