Finite Group Databasebeta

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C2×#10710959 #72248024

order1536
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#10710959

Splittings

FittingC2×C3×#2134 ⋊ C2
SylowC3 ⋊ C2×#12755

Subgroups

subgroupG/H
centerC22#70033
commutatorC2×C3×D4C23×C4
FrattiniC2×#41C23×D3
FittingC2×C3×#2134C2
socleC22×C3#1790
prime radicalC2×#86334C2
cosocleC2×C3×#41C24
hypercenterC2×#2134D3
hypocenterC3C2×#12755
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#2134

5598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72248059, #72254214, C2×D3×#12755
maximal normal subgroup of#72248059, #72254214, C2×D3×#12755
maximal quotient of#72271669

GAP

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

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