Finite Group Databasebeta

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Cayley diagram

C8×C49 #87622

order392
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C23×C3×C7 (order 168) · Inn(G) = C1 · Out(G) = C23×C3×C7
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC392
abelianC8×C49

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC4×C7C2×C7
socle, prime radicalC2×C7C4×C7
LCSG ▷ C1
UCSC1 ◁ G

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 392

A minimal faithful permutation representation — 1 generator on 57 points:

(1,2,3,4,5,6,7,8)(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)

Conjugacy classes & character table

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

Relationships

center ofM32×C49, #131×C49
derived subgroup of#13095839, D784, Dic392, …
Frattini subgroup ofDic8232, D16464, Dic13720, …
Fitting subgroup ofC49⋊C16, Dic196, C8×D49, …
maximal subgroup ofC49⋊C16, C16×C49, Dic196, …
maximal normal subgroup ofC49⋊C16, C16×C49, Dic196, …
maximal quotient ofC16×C49, C2×C8×C49, C3⋊C8×C49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(392, 2);