C₇×#7478 #13184694

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7478

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#357
commutator	C₁₆	C₄²×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

174 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 2 generators on 43 points:

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

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, 466);