Finite Group Databasebeta

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Cayley diagram

C3×C4×D8 #5311

order192
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C23×#3003 (order 1024) · Inn(G) = D4 · Out(G) = C24×D4
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×C4×D8

Subgroups

subgroupG/H
centerC2×C3×C4D4
commutatorC4C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C2×D4
prime radicalC2×C3×D8C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ G

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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67762, #67763, D8×C3⋊C8, …
maximal subgroup of#67762, #67763, C3×#1331, …
maximal normal subgroup of#67762, #67763, C3×#1331, …
maximal quotient ofC3×C8×D8, C3×#1581, C3×#1589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 870);