A₄×#6909 #72228401

order	2916
automorphism	?
homology	Abelianization (H₁) = C₃⁴ · Schur multiplier (H₂) = C₂×C₃⁶

Aliases

direct product

A₄×#6909

Splittings

Fitting	C₂²×#6909 ⋊ C₃
Sylow	C₂² ⋊ C₃×#6909

Subgroups

	subgroup	G/H
center	C₃	C₃×A₄×H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃⁴
Frattini	C₃²	C₃³×A₄
Fitting	C₂²×#6909	C₃
socle	C₂²×C₃	C₃²×H(3,𝔽₃)
hypercenter	#6909	A₄
hypocenter	C₂²	C₃×#6909
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6909

1270 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 271 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) = 108

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×#10624749, A₄×#4183⋊C₃, S₄×#6909, …
maximal normal subgroup of	A₄×#10624749, A₄×#4183⋊C₃, S₄×#6909, …

GAP

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

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