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Cayley diagram

C22×C97 #87606

order388
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C3×D3×C32 (order 576) · Inn(G) = C1 · Out(G) = C3×D3×C32
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC22×C97

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 101
transitive degreeμt(G) = 388

A minimal faithful permutation representation — 2 generators on 101 points:

(1,2)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(3,4)

Conjugacy classes & character table

388 conjugacy classes. The character table is too large to display.

Relationships

center ofC22⋊C4×C97, C4⋊C4×C97, C2×D4×C97, …
derived subgroup ofC22⋊(C97⋊C3), #15690724, #15690725, …
Fitting subgroup ofC2×Dic97, C22⋊D97, C22×D97, …
socle ofC2×Dic97, C2×C4×C97, C22×D97, …
maximal subgroup ofC2×Dic97, C22⋊D97, C2×C4×C97, …
maximal normal subgroup ofC2×Dic97, C22⋊D97, C2×C4×C97, …
maximal quotient ofC2×C4×C97, D4×C97, Q8×C97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(388, 5);