C₇×#12021 #13189237

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#12021

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

216 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 47
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,10,13,11,12,14,15)(16,17)(18,27,30,42)(19,24)(20,33)(21,36)(22,26)(23,29,38,45)(25,34)(28,41,31,43)(32,40)(35,47)(37,44,39,46);(16,18,20,29,21,28,34,44,22,30,35,45,19,31,32,46)(17,23,25,27,24,39,33,43,26,38,40,42,36,37,47,41);(9,12)(13,15)(16,19,22,21)(17,24,26,36)(18,28,30,31)(20,32,35,34)(23,37,38,39)(25,33,40,47)(27,41,42,43)(29,44,45,46)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5009);