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Cayley diagram

C2×#362 #2147

order128
nilpotency class4
isoclinismisoclinic class of #360
automorphismAut(G) order 2048
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#362

Subgroups

subgroupG/H
center, socleC22#102
commutatorC8C22×C4
Frattini, cosocleC2×C8C23
prime radicalC23SD16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

108 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);(3,4,7,11,8,13,23,12)(5,14,16,27,17,31,22,30)(6,18,20,10,21,28,34,29)(9,19,24,15,26,32,25,33);(3,5,8,17)(4,9,13,26)(6,19,21,32)(7,22,23,16)(10,27,29,30)(11,25,12,24)(14,18,31,28)(15,20,33,34)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#362, #12648, #12737, …
maximal normal subgroup ofC4×#362, #12648, #12737, …
maximal quotient ofC2×#1334, #11600, #11653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 879);