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C2×#87704 #11716331

order798
isoclinismisoclinic class of #87704
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×#87704

Splittings

Fitting, socle, HallC2×C7×C19 ⋊ C3
derived, HallC7×C19 ⋊ C2×C3
HallC2×C19 ⋊ ΓL(1,𝔽8)
SylowC19 ⋊ C2×ΓL(1,𝔽8)
HallC2×C7 ⋊ C19⋊C3
SylowC7 ⋊ C2×C19⋊C3

Subgroups

subgroupG/H
center, hypercenterC2#87704
commutator, cosocle, hypocenterC7×C19C2×C3
FrattiniC1G
Fitting, socleC2×C7×C19C3
LCSG ▷ C7×C19
UCSC1 ◁ C2

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 266

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13097582, #13097586, #13097588, …
maximal normal subgroup of#13097582, #13097586, #13097588, …
maximal quotient ofC4×#87704, C22×#87704, C2×#11924776, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(798, 15);