C₇×#8801 #13186017

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#8801

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#383
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

306 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 1789);