C₇×#7466 #13184682

order	1792
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7466

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₄²
commutator	C₄	C₄×C₇×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂²×C₇	#346
prime radical	C₂³×C₇	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

182 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 118 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)(12,13,15,21,16,23,27,36,17,19,28,31,29,32,40,41,18,22,26,35,30,37,39,43,14,20,24,33,25,34,38,42);(9,11)(13,19,22,20)(15,26)(21,33,35,31)(23,32,37,34)(24,28)(27,39)(36,42,43,41)(38,40)

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, 454);