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Cayley diagram

C2×#359 #2142

order128
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C2×#10576928 (order 1024) · Inn(G) = D8 · Out(G) = C22×#41
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#359

Subgroups

subgroupG/H
centerC2×C4D8
commutatorC8C22×C4
Frattini, cosocleC2×C8C23
socleC22#102
prime radicalC2×#135C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

112 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)(5,14,16,30,17,31,24,28)(6,18,20,29,22,33,21,11)(10,19,25,32,27,34,26,15);(3,5)(4,10)(6,19)(7,16)(8,24)(9,17)(11,28)(12,25)(13,27)(14,18)(15,21)(20,32)(22,34)(23,26)(29,30)(31,33)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#359, #12640, #12641, …
maximal normal subgroup ofC4×#359, #12640, #12641, …
maximal quotient ofC2×#1331, C2×#1332, #11591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 874);