AGL(1,𝔽₇)×GU(2,𝔽₃) #72246272

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

AGL(1,𝔽₇)×GU(2,𝔽₃)

Splittings

derived	C₇×SL(2,𝔽₃) ⋊ C₂×C₃×C₄
Sylow	C₇ ⋊ C₂×C₃×GU(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	S₄×AGL(1,𝔽₇)
commutator, hypocenter	C₇×SL(2,𝔽₃)	C₂×C₃×C₄
Fitting	C₇×#41	C₂×C₃×D₃
socle	C₂×C₇	C₂×C₃×SL(2,ℤ/4)
prime radical	AGL(1,𝔽₇)×#41⋊C₃	C₂
cosocle	C₇×#41⋊C₃	C₂²×C₃
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₄

350 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 31 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#5429
maximal normal subgroup of	AGL(1,𝔽₇)×#5429

GAP

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

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