C₂×D₃×#5932 #72249244

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×D₃×#5932

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×#1020
commutator, cosocle, hypocenter	C₃×#997	C₂³
Frattini	C₂	C₂×D₃×#1020
Fitting	C₂×C₃×H(5,𝔽₂)	C₂×D₃
socle	C₂²×C₃	C₂×#1020
prime radical	C₂×D₃×#997	C₂
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂²

3457 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 3457 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Conjugacy classes & character table

78 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((5,13,11)(6,17,12)(8,16,14)(9,18,15)(20,21), (1,2)(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,17)(16,18)(19,20,21), (3,4,7,10)(5,9,12,16)(6,8,13,15)(11,14,17,18));