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C2×C4×#3861 #11924259

order1176
isoclinismisoclinic class of #3861
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×C4×#3861

Splittings

Fitting, HallC2×C4×C72 ⋊ C3
derived, SylowC72 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4#3861
commutator, hypocenterC72C2×C3×C4
FrattiniC2C22×#3861
FittingC2×C4×C72C3
socleC22×C72C2×C3
prime radicalC22×#3861C2
cosocleC2×C72C22×C3
LCSG ▷ C72
UCSC1 ◁ C2×C4

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 392

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×#3861, C2×C4⋊#64424, C2×C4×#64424, …
maximal normal subgroup ofC2×D4×#3861, C2×C4⋊#64424, C2×C4×#64424, …
maximal quotient ofC22⋊C4×#3861, C42×#3861, C2×C8×#3861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 174);