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Cayley diagram

C2×#117 #514

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 8192
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#117

Subgroups

subgroupG/H
centerC22×C4C22
commutatorC2C23×C4
Frattini, cosocleC22C24
socleC23C23
prime radicalC24C22
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22 ▷ C1

165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#142224, #142225, #142226, …
Frattini subgroup of#7188, #7190, #7214, …
maximal subgroup of#1458, #1464, #1729, …
maximal normal subgroup of#1458, #1464, #1729, …
maximal quotient ofC2×#376, C2×C4×C22⋊C4, C2×C4×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 195);