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C4×C5×C59 #11924375

order1180
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C4×C29 (order 464) · Inn(G) = C1 · Out(G) = C22×C4×C29
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC1180
abelianC4×C5×C59
multiplicative(ℤ/1181)ˣ, (ℤ/2362)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C2×C5×C59
socle, prime radicalC2×C5×C59C2
LCSG ▷ C1
UCSC1 ◁ G

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 1180

A minimal faithful permutation representation — 1 generator on 68 points:

(1,2,3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)

Conjugacy classes & character table

1180 conjugacy classes. The character table is too large to display.

Relationships

center ofC4×C5×ΓL(1,𝔽8)×C59
Fitting subgroup ofC4×C5×C59⋊C29
maximal subgroup ofC3×C4×C5×C59, C5×Dic3×C59, C4×C5×C11×C59, …
maximal normal subgroup ofC3×C4×C5×C59, C4×C5×C11×C59, C4×C5×C13×C59, …
maximal quotient ofC3×C4×C5×C59, C5×Dic3×C59, C4×C5×C11×C59, …
automorphism group ofC1181, C2×C1181
outer automorphism group ofC1181, C2×C1181

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1180, 4);