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Cayley diagram

C22×C4⋊C4 #513

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 49152
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×C4⋊C4

Subgroups

subgroupG/H
center, socle, prime radicalC24C22
commutatorC2C23×C4
Frattini, cosocleC22C24
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C22 ▷ C1

209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 12 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#142220, #142221, #142222, …
Frattini subgroup of#7189, #7215, #7251, …
Fitting subgroup ofA4×C4⋊C4, #73050, #73051, …
maximal subgroup ofC4⋊#95, #1436, #1439, …
maximal normal subgroup ofC4⋊#95, #1436, #1439, …
maximal quotient ofC22×#95, C2×C4×C4⋊C4, C2×#380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 194);