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Cayley diagram

C3×C4×Q8 #959

order96
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C23×#1020 (order 768) · Inn(G) = C22 · Out(G) = C23×S4
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×C4×Q8

Subgroups

subgroupG/H
centerC2×C3×C4C22
commutatorC2C22×C3×C4
Frattini, cosocleC22C23×C3
socle, prime radicalC22×C3C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

70 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)(13,14,15);(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#79911⋊C3, #79910⋊C3, #79914⋊C3, …
Fitting subgroup of#4535, #5020, #5021, …
maximal subgroup of#4535, C3×Q8⋊C8, #5020, …
maximal normal subgroup of#4535, C3×Q8⋊C8, #5020, …
maximal quotient ofC3×C4×C4⋊C4, C3×#387, C3×#389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 166);