Finite Group Databasebeta

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Cayley diagram

C2×C3×C4⋊C4 #956

order96
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#347042 (order 1024) · Inn(G) = C22 · Out(G) = C22×B(4,𝔽2)
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×C4⋊C4

Subgroups

subgroupG/H
center, socle, prime radicalC23×C3C22
commutatorC2C22×C3×C4
Frattini, cosocleC22C23×C3
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10682711, #10682712, #10682713, …
Frattini subgroup ofC9×#1318, C9×#1319, C9×#1324, …
Fitting subgroup of#4528, #4529, #4530, …
maximal subgroup of#4528, #4529, #4530, …
maximal normal subgroup of#4528, #4529, #4530, …
maximal quotient ofC2×C3×#95, C3×C4×C4⋊C4, C3×#380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 163);