C₇×#8810 #13186026

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#8810

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

402 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 35
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1798);