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C2×C3×#14524189 #27188563

order12978
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×#14524189

Splittings

Fitting, socleC2×C3×C7×C103 ⋊ C3
derived, HallC7×C103 ⋊ C2×C32
HallC2×C7×C103 ⋊ C32
HallC2×C103 ⋊ C3×ΓL(1,𝔽8)
SylowC103 ⋊ C2×C3×ΓL(1,𝔽8)
HallC2×C7 ⋊ C3×C103⋊C3
SylowC7 ⋊ C2×C3×C103⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C3#14524189
commutator, cosocle, hypocenterC7×C103C2×C32
FrattiniC1G
Fitting, socleC2×C3×C7×C103C3
LCSG ▷ C7×C103
UCSC1 ◁ C2×C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 115
transitive degreeμt(G) = 4326

A minimal faithful permutation representation — 2 generators on 115 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,69)(41,56,66)(44,59,64)(47,62,61)(50,54,58)(52,67,55)(57,72,84)(60,75,82)(63,78,80)(65,73,77)(68,83,74)(70,85,71)(76,88,95)(79,90,94)(81,89,92)(86,96,87)(91,97,101)(93,98,100)(99,102,103)(105,106,107)(108,110,109)(114,115);(1,2,4,8,15,25,38,54,33,12,19,29,42,58,74,87,96,67,35,48,64,80,92,72,37,52,68,69,84,95,101,103,97,75,40,14,23,34,47,63,79,91,98,78,43,17,3,6,11,18,28,41,57,73,46,20,5,9,16,26,39,55,71,83,49,22,32,45,61,77,53,70,86,85,51,66,82,94,100,88,56,24,36,50,65,81,93,99,102,90,59,27,7,13,21,31,44,60,76,89,62,30,10)(104,105,106,108,107,109,110)(111,112,113)

Conjugacy classes & character table

1458 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×#22575751, C3×#22575747, C3×#22575743, …
maximal normal subgroup ofC3×#22575751, C3×#22575747, C3×#22575743, …
maximal quotient ofC3×C4×#14524189, C22×C3×#14524189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12978, 58);