C₂×D₃×#87519 #72249259

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×D₃×#87519

Splittings

Fitting

C₂×C₃×#585 ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₃×#1020
commutator, cosocle, hypocenter	C₃×#997	C₂⁴
Frattini	C₂	C₂²×D₃×#1020
Fitting	C₂×C₃×#585	C₂×D₃
socle	C₂²×C₃	C₂²×#1020
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂×C₄

27777 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 27777 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 21 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

automorphism group of

C₂²×C₃⋊C₁₆

GAP

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

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