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Cayley diagram

D4×C3⋊C8 #5010

order192
automorphismAut(G) = C23×D3×#120 (order 1536) · Inn(G) = C22×D3 · Out(G) = C23×D4
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productD4×C3⋊C8

Splittings

SylowC3 ⋊ C8×D4

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×D3
commutatorC2×C3C22×C8
FittingC3×C4×D4C2
socleC22×C3C22×C4
prime radicalC2×C3×D4C4
cosocleC2×C3×C4C23
hypercenterC4×D4D3
hypocenterC3C8×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×D4

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#68148, #68149, #68150, …
maximal normal subgroup of#68148, #68149, #68150, …
maximal quotient ofC22⋊C4×C3⋊C8, #69449, #69451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 569);