C₇×#11917 #13189133

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11917

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1533
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#97	C₂³×C₇
prime radical	C₇×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 280 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)(8,9,12,16,11,17,24,35,13,18,28,36,25,15,27,33)(10,19,21,37,22,14,29,31,23,30,38,34,20,32,26,39);(8,10)(9,14)(11,20)(12,26)(13,23)(15,30)(16,34)(17,19)(18,32)(21,27)(22,25)(24,38)(28,29)(31,35)(33,39)(36,37);(8,11,13,25)(9,15,18,17)(10,20,23,22)(12,27,28,24)(14,30,32,19)(16,35,36,33)(21,29,38,26)(31,37,39,34)

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, 4905);