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Cayley diagram

C2×#904 #5741

order192
isoclinismisoclinic class of #204
automorphismAut(G) = D3×#72214076 (order 6144) · Inn(G) = D12 · Out(G) = C2×#2403
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#904

Splittings

FittingC22×C3×C8 ⋊ C2
SylowC3 ⋊ C2×#135

Subgroups

subgroupG/H
centerC2×C4D12
commutator, cosocleC3×C4C24
FrattiniC4C23×D3
FittingC22×C3×C8C2
socleC22×C3C2×D4
hypercenterC22×C8D3
hypocenterC3C2×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

266 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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#68695, #68946, #69109, …
maximal normal subgroup of#68695, #68946, #69109, …
maximal quotient ofC2×#4683, #78442, C2×C4×D24, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1300);