C₂×D₃×#5950 #72249237

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×D₃×#5950

Splittings

Fitting

C₂×C₃×#585 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₃×C₂²⋊A₄
commutator, cosocle, hypocenter	C₃×H(5,𝔽₂)	C₂³×C₃
Frattini	C₂	C₂²×D₃×C₂²⋊A₄
Fitting	C₂×C₃×#585	C₂×C₃
socle	C₂²×C₃	C₂²×C₂²⋊A₄
LCS	G ▷ C₃×H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₄

20689 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 20689 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

(6,7,13)(9,10,16)(11,12,17)(14,15,18)(20,21);(3,5,8,4)(6,9,14,11)(7,10,15,12)(13,17,18,16)(19,20,21);(1,2)(3,16,8,17)(4,18,5,13)(6,10,14,12)(7,9,15,11)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#87519
maximal normal subgroup of	C₂×D₃×#87519

GAP

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

gap> G := Group((6,7,13)(9,10,16)(11,12,17)(14,15,18)(20,21), (3,5,8,4)(6,9,14,11)(7,10,15,12)(13,17,18,16)(19,20,21), (1,2)(3,16,8,17)(4,18,5,13)(6,10,14,12)(7,9,15,11));