C₇×#7851 #13185067

order	1792
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#7851

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₇	C₂³
commutator	C₂³	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂³×C₇	C₂×C₄²
prime radical	C₂⁴×C₇	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₇ ◁ G

610 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 43
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 839);