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C25×C29 #10624385

order725
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C42×C5×C7 (order 560) · Inn(G) = C1 · Out(G) = C42×C5×C7
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC725
abelianC25×C29

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC5C5×C29
socle, prime radicalC5×C29C5
LCSG ▷ C1
UCSC1 ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 725

A minimal faithful permutation representation — 1 generator on 54 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)

Conjugacy classes & character table

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

Relationships

center ofD3×C25×C29, ΓL(1,𝔽8)×C25×C29, D11×C25×C29, …
derived subgroup ofD725, Dic725, #15668502, …
Fitting subgroup ofC29×D25, C25×D29, D725, …
maximal subgroup ofC29×D25, C25×D29, D725, …
maximal normal subgroup ofC29×D25, C25×D29, D725, …
maximal quotient ofC2×C25×C29, C3×C25×C29, C29×C125, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(725, 1);