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Cayley diagram

C2×#95 #375

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = #72255861 (order 12288) · Inn(G) = C22 · Out(G) = C2×#72209884
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C25

Aliases

direct productC2×#95

Subgroups

subgroupG/H
center, socle, prime radicalC24C22
commutatorC2C2×C42
Frattini, cosocleC23C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#93213, #93214, #93215, …
Frattini subgroup of#7103, #7104, #7122, …
Fitting subgroup ofC2×#95⋊C3, SL(2,ℤ/8), C2×#4621, …
maximal subgroup of#1280, #1294, C4×#95, …
maximal normal subgroup of#1280, #1294, C4×#95, …
maximal quotient ofC4×#95, C22⋊#95, C4⋊#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 56);