C₂×#11923643 #72249243

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

Aliases

direct product

C₂×#11923643

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
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂²

3461 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 3461 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

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