Finite Group Databasebeta

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C2×#72214133 #72229337

order5408
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#72214133

Splittings

SylowC132 ⋊ C2×C4⋊C4

Subgroups

subgroupG/H
center, hypercenterC22D13⋊#261
commutatorC2×C132C22×C4
FrattiniC2C2×D13⋊#261
Fitting, socleC22×C132C2×C4
prime radicalC22×C13⋊D13C22
cosocleC2×C13⋊D13C23
hypocenterC132C2×C4⋊C4
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C22

346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 104

A minimal faithful permutation representation — 3 generators on 32 points:

(4,6)(7,17,15,18)(8,11,19,13)(9,16,14,12)(20,21,29,24)(23,26,31,32)(25,30,28,27);(1,2);(3,4)(5,6)(8,9,10,12)(11,14,15,19)(13,17,18,16)(21,23,25,24)(22,27,26,28)(29,30,31,32)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72214141, C2×#72214198, #72259321
maximal normal subgroup ofC2×#72214141, C2×#72214198, #72259321
maximal quotient of#72259282

GAP

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

gap> G := Group((4,6)(7,17,15,18)(8,11,19,13)(9,16,14,12)(20,21,29,24)(23,26,31,32)(25,30,28,27), (1,2), (3,4)(5,6)(8,9,10,12)(11,14,15,19)(13,17,18,16)(21,23,25,24)(22,27,26,28)(29,30,31,32));