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Cayley diagram

C2×#568 #3572

order128
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) order 49152
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#568

Subgroups

subgroupG/H
centerC2×C4C24
commutatorC2C24×C4
Frattini, cosocleC4C25
socleC22C23×C4
prime radicalC22×#41C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4,6,10,7,9,13,16)(5,8,12,15,11,14,17,18);(3,5)(4,8)(6,12)(7,11)(9,14)(10,15)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18);(4,9)(8,14)(10,16)(15,18)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#568⋊C3, #11711310, #11711312, …
maximal subgroup of#10742, #10744, #10746, …
maximal normal subgroup of#10742, #10744, #10746, …
maximal quotient ofC2×#2873, C2×#2875, C4×#568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2304);