C₂×C₇×#1407 #13190792

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₇×#1407

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#353
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#124 ◁ G

416 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 41
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6564);