AGL(1,𝔽₇)×#5429 #72246277

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

AGL(1,𝔽₇)×#5429

Splittings

Fitting	C₇×H(5,𝔽₂) ⋊ C₂×C₃×D₃
Sylow	C₇ ⋊ C₂×C₃×#5429

Subgroups

	subgroup	G/H
center	C₂	AGL(1,𝔽₇)×GL(2,ℤ/4)
commutator, cosocle	C₇×#41⋊C₃	C₂³×C₃
Frattini	C₄	C₂×S₄×AGL(1,𝔽₇)
Fitting	C₇×H(5,𝔽₂)	C₂×C₃×D₃
socle	C₂×C₇	C₂×C₃×GL(2,ℤ/4)
hypercenter	Q₈	S₄×AGL(1,𝔽₇)
hypocenter	C₇×SL(2,𝔽₃)	C₂×C₃×D₄
LCS	G ▷ C₇×#41⋊C₃ ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

1450 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1450 conjugacy classes of subgroups — too many to draw. Only its 89 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:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

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