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Cayley diagram

#1181

order116
isoclinismisoclinic class of #1181
automorphismAut(G) = AGL(1,𝔽29) (order 812) · Inn(G) = G · Out(G) = C7
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, SylowC29 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC29C4
prime radical, cosocleD29C2
LCSG ▷ C29
UCSC1

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 29

A minimal faithful permutation representation — 2 generators on 29 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,25)(14,21,22,23)(16,24,19,20)(26,28,29,27);(1,2,5,10,19,15,23,17,26,28,24,12,3,7,14,22,13,6,11,20,27,29,25,21,8,16,18,9,4)

Polycyclic presentation:

2^2,29|a^2=b,c^a=c^12,c^b=c^28

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9C10C11
size14444444292929
order129292929292929244
χ111111111111
χ2111111111-1-1
χ311111111-1i−i
χ411111111-1−ii
χ54−ζ10−ζ54−ζ62−ζ106−ζ6−ζ14−ζ102−ζ110−ζ38−ζ50−ζ66−ζ78−ζ30−ζ46−ζ70−ζ86−ζ18−ζ42−ζ74−ζ98−ζ22−ζ26−ζ90−ζ94−ζ2−ζ34−ζ82−ζ114000
χ64−ζ6−ζ14−ζ102−ζ110−ζ38−ζ50−ζ66−ζ78−ζ30−ζ46−ζ70−ζ86−ζ18−ζ42−ζ74−ζ98−ζ2−ζ34−ζ82−ζ114−ζ10−ζ54−ζ62−ζ106−ζ22−ζ26−ζ90−ζ94000
χ74−ζ38−ζ50−ζ66−ζ78−ζ30−ζ46−ζ70−ζ86−ζ18−ζ42−ζ74−ζ98−ζ2−ζ34−ζ82−ζ114−ζ22−ζ26−ζ90−ζ94−ζ6−ζ14−ζ102−ζ110−ζ10−ζ54−ζ62−ζ106000
χ84−ζ30−ζ46−ζ70−ζ86−ζ18−ζ42−ζ74−ζ98−ζ2−ζ34−ζ82−ζ114−ζ22−ζ26−ζ90−ζ94−ζ10−ζ54−ζ62−ζ106−ζ38−ζ50−ζ66−ζ78−ζ6−ζ14−ζ102−ζ110000
χ94−ζ18−ζ42−ζ74−ζ98−ζ2−ζ34−ζ82−ζ114−ζ22−ζ26−ζ90−ζ94−ζ10−ζ54−ζ62−ζ106−ζ6−ζ14−ζ102−ζ110−ζ30−ζ46−ζ70−ζ86−ζ38−ζ50−ζ66−ζ78000
χ104−ζ22−ζ26−ζ90−ζ94−ζ10−ζ54−ζ62−ζ106−ζ6−ζ14−ζ102−ζ110−ζ38−ζ50−ζ66−ζ78−ζ30−ζ46−ζ70−ζ86−ζ2−ζ34−ζ82−ζ114−ζ18−ζ42−ζ74−ζ98000
χ114−ζ2−ζ34−ζ82−ζ114−ζ22−ζ26−ζ90−ζ94−ζ10−ζ54−ζ62−ζ106−ζ6−ζ14−ζ102−ζ110−ζ38−ζ50−ζ66−ζ78−ζ18−ζ42−ζ74−ζ98−ζ30−ζ46−ζ70−ζ86000

Here ζ is a primitive root of unity of order 116 (ζ116 = 1); i = ζ29.

Relationships

maximal subgroup ofC2×#1181, C3×#1181, C3⋊#1181, …
maximal normal subgroup ofC2×#1181, C3×#1181, C5×#1181, …
maximal quotient of#6599, C2×#1181, C3×#1181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(116, 3);