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Cayley diagram

C2×#368 #2160

order128
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C2×#72213937 (order 2048) · Inn(G) = D8 · Out(G) = C2×#525
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×#368

Subgroups

subgroupG/H
centerC2×C4D8
commutatorC8C22×C4
Frattini, cosocleC2×C8C23
socleC22#107
prime radicalC23Q16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2);(3,4,7,12,9,13,8,23)(5,14,17,29,19,30,18,33)(6,20,21,27,22,34,15,11)(10,24,25,32,26,31,28,16);(3,5,15,31,8,18,22,32,9,19,21,24,7,17,6,16)(4,10,20,29,12,25,27,30,13,26,34,33,23,28,11,14)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#7472, #7473, #7474, …
maximal normal subgroup of#7472, #7473, #7474, …
maximal quotient ofC2×#1371, C2×#1372, #12008, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 892);