Finite Group Databasebeta

← Prev (#14515365) Next (#14515367) →

ΓL(1,𝔽8)×Dic23 #14515366

order1932
isoclinismisoclinic class of ΓL(1,𝔽8)×D23
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×Dic23
unambiguous semidirect productC23⋊(C4×ΓL(1,𝔽8))
unambiguous semidirect product(C7×Dic23)⋊C3

Splittings

derived, HallC7×C23 ⋊ C3×C4
HallC7×Dic23 ⋊ C3
HallΓL(1,𝔽8)×C23 ⋊ C4
SylowC23 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×Dic23

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D23
commutator, hypocenterC7×C23C3×C4
Fitting, socle, cosocleC2×C7×C23C2×C3
prime radicalC2×ΓL(1,𝔽8)×C23C2
LCSG ▷ C7×C23
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 644

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3,4)(5,7,6)(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);(1,2,3,5,4,6,7)(8,10,9,11)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic23×C49⋊C3, C7×ΓL(1,𝔽8)×Dic23, C23⋊#10596873, …
maximal normal subgroup ofC7×ΓL(1,𝔽8)×Dic23, C11×ΓL(1,𝔽8)×Dic23, ΓL(1,𝔽8)×Dic23⋊C11, …
maximal quotient ofDic23×C49⋊C3, C7×ΓL(1,𝔽8)×Dic23, C23⋊#10596873, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1932, 1);