D₇×#5465 #72277214

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

D₇×#5465

Splittings

Fitting, derived	C₇×H(3,𝔽₄) ⋊ C₂×C₃
Hall	D₇×H(3,𝔽₄) ⋊ C₃
Sylow	C₇ ⋊ C₂×#5465

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₇×C₂²⋊A₄
commutator, Fitting, cosocle, hypocenter	C₇×H(3,𝔽₄)	C₂×C₃
socle	C₂²×C₇	C₂×C₂²⋊A₄
LCS	G ▷ C₇×H(3,𝔽₄)
UCS	C₁ ◁ C₂²

686 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(2,3)(4,5)(6,7)(9,10,12)(13,14,15)(17,18,20)(21,22,23);(1,2,4,6,7,5,3)(9,13)(12,15)(17,21)(18,22);(8,12)(9,10)(11,15)(13,14)(16,20)(17,18)(19,23)(21,22)

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

D₇×#11710730

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3)(4,5)(6,7)(9,10,12)(13,14,15)(17,18,20)(21,22,23), (1,2,4,6,7,5,3)(9,13)(12,15)(17,21)(18,22), (8,12)(9,10)(11,15)(13,14)(16,20)(17,18)(19,23)(21,22));