C₇×#12858 #13190074

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#12858

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×D₄	C₂³×C₇
prime radical	C₂³×C₇×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁴×C₇ ◁ C₂³×C₇×D₄ ◁ G

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

normal subgroups only

This group has 786 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)(26,28,31,30);(9,12)(13,15)(16,18)(17,21)(19,26)(20,25)(22,30)(23,28)(24,29)(27,31);(8,10,11,14)(9,13,12,15)(17,22)(18,25)(19,27)(21,28)(23,24)(29,30)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5846);