D₇×SO⁺(4,𝔽₃) #72246263

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

D₇×SO⁺(4,𝔽₃)

Splittings

Fitting	C₇×H(5,𝔽₂) ⋊ C₂×C₃⋊D₃
derived	C₇×Ω⁺(4,𝔽₃) ⋊ C₂²
Sylow	C₇ ⋊ C₂×SO⁺(4,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇×A₄⋊S₄
commutator, cosocle, hypocenter	C₇×Ω⁺(4,𝔽₃)	C₂²
Fitting	C₇×H(5,𝔽₂)	C₂×C₃⋊D₃
socle	C₂×C₇	C₂×A₄⋊S₄
LCS	G ▷ C₇×Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₂

934 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 168

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

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×SO⁺(4,𝔽₃)
maximal normal subgroup of	AGL(1,𝔽₇)×SO⁺(4,𝔽₃)

GAP

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

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