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C2×#11924148 #72228583

order2352
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×#11924148

Splittings

derived, SylowC72 ⋊ C2×C3×C8
HallC2×C7⋊(C7⋊C8) ⋊ C3
Hall#3861 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2×C4#64424
commutator, hypocenterC72C2×C3×C8
FrattiniC4C2×#64424
FittingC2×C4×C72C2×C3
socleC22×C72C3×C4
prime radicalC22×#3861C4
cosocleC4×C72C22×C3
LCSG ▷ C72
UCSC1 ◁ C2×C4

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 784

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(12,14,15)(13,16,17)(18,19,24)(20,21,23);(3,4,5,7,6,8,9,10)(12,13)(14,16)(15,17)(19,20)(21,23)(22,24);(11,12,15,16,14,17,13)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C8×#64424, #72228589, #72228642, …
maximal normal subgroup ofC2×C8×#64424, #72228589, #72228642, …

GAP

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

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