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Cayley diagram

C2×#1181 #6608

order232
isoclinismisoclinic class of #1181
automorphismAut(G) = C2×AGL(1,𝔽29) (order 1624) · Inn(G) = #1181 · Out(G) = C2×C7
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×#1181
unambiguous semidirect productD29⋊C4

Splittings

Fitting, socleC2×C29 ⋊ C4
derived, SylowC29 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2#1181
commutator, hypocenterC29C2×C4
FrattiniC1G
Fitting, socleC2×C29C4
prime radicalC2×D29C2
cosocleD29C22
LCSG ▷ C29
UCSC1 ◁ C2

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 58

A minimal faithful permutation representation — 2 generators on 31 points:

(4,5,6,8)(7,10,11,17)(9,14,15,13)(12,19,20,27)(16,23,24,25)(18,26,21,22)(28,30,31,29);(1,2)(3,4,7,12,21,17,25,19,28,30,26,14,5,9,16,24,15,8,13,22,29,31,27,23,10,18,20,11,6)

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#1181, C4⋊#1181, C22⋊#1181, …
maximal normal subgroup ofC4×#1181, C4⋊#1181, C22⋊#1181, …
maximal quotient ofD29⋊C8, #90796, C4×#1181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(232, 12);