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C2×#13103978 #72226383

order3200
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#13103978

Splittings

SylowC52 ⋊ C2×C4×C4⋊C4

Subgroups

subgroupG/H
centerC23C2×D5⋊AGL(1,𝔽5)
commutatorC2×C52C22×C42
FrattiniC22C22×D5⋊AGL(1,𝔽5)
FittingC22×C4×C52C2×C4
socleC23×C52C22×C4
prime radicalC23×C5⋊D5C23
cosocleC22×C5⋊D5C24
hypercenterC22×C4D5⋊AGL(1,𝔽5)
hypocenterC52C2×C4×C4⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

2354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×#11717363
maximal normal subgroup ofC2×C4×#11717363
maximal quotient of#72226385, #72226386

GAP

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

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