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Cayley diagram

C4×D3×Q8 #5571

order192
automorphismAut(G) order 9216
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×D3×Q8

Splittings

FittingC3×C4×Q8 ⋊ C2
SylowC3 ⋊ C2×C4×Q8

Subgroups

subgroupG/H
centerC2×C4C22×D3
commutatorC2×C3C23×C4
FrattiniC22C23×D3
FittingC3×C4×Q8C2
socle, cosocleC22×C3C24
prime radicalC22×D3C23
hypercenterC4×Q8D3
hypocenterC3C2×C4×Q8
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×Q8

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

(5,7,8,12)(6,9,10,11)(13,14,15);(5,6,8,10)(7,11,12,9);(14,15);(1,2,3,4)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×Q8⋊C8, #68231, #68236, …
maximal normal subgroup ofD3×Q8⋊C8, #68231, #68236, …
maximal quotient ofC4×#887, C4×#889, C4×#888, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1130);