C₇×#10083 #13187299

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10083

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

526 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 114 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)(12,13,17,21)(14,22,24,27)(15,20,16,19)(18,25,26,23)(28,29,32,35)(30,36,38,43)(31,39,41,34)(33,40,42,37);(8,9)(10,11)(12,14)(13,18)(15,25)(16,23)(17,24)(19,27)(20,22)(21,26)(28,30,32,38)(29,33,35,42)(31,40,41,37)(34,43,39,36);(9,11)(12,14)(13,18)(15,23)(16,25)(17,24)(19,22)(20,27)(21,26)(31,41)(34,39)(36,43)(37,40)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3071);