C₂×C₃⋊SO⁺(4,𝔽₃) #72242404

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²×C₃³

Aliases

direct product

C₂×C₃⋊SO⁺(4,𝔽₃)

Splittings

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

Subgroups

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

1582 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 1582 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 29
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×SO⁺(4,𝔽₃), C₂×#72220371, #72252919
maximal normal subgroup of	C₂×D₃×SO⁺(4,𝔽₃), C₂×#72220371, #72252919
maximal quotient of	C₂×#72220369

GAP

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

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