C₇×#8139 #13185355

order	1792
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#8139

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

330 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 448

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)(18,26,29,39)(19,23)(20,24)(21,25)(22,28,34,42)(27,38,41,46)(30,35)(31,36)(32,37)(33,40,44,47)(43,45);(16,18)(17,22,25,34)(19,27)(20,28)(21,29)(23,33,36,44)(24,26,37,39)(30,40)(31,41)(32,42)(35,38,45,46)(43,47);(9,12)(13,15)(16,19,20,30,21,31,32,43)(17,23,24,35,25,36,37,45)(18,27,28,40,29,41,42,47)(22,33,26,38,34,44,39,46)

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1127);