C₇×#12857 #13190073

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#12857

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×D₄	C₂³×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×#41 ◁ G

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

normal subgroups only

This group has 670 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) = 31
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5845);