C₇×#7474 #13184690

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1426
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7474

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#367
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) = 448

A minimal faithful permutation representation — 2 generators on 43 points:

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

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, 462);