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Cayley diagram

C2×#1007 #5960

order192
automorphismAut(G) order 73728
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#1007

Splittings

FittingC22×C3×Q8 ⋊ C2
SylowC3 ⋊ C2×2−1+4

Subgroups

subgroupG/H
centerC22C23×D3
commutator, cosocleC2×C3C25
FrattiniC2C24×D3
FittingC22×C3×Q8C2
socleC22×C3C24
hypercenterC22×Q8D3
hypocenterC3C2×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×Q8

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 21 points:

(1,2)(3,4,5)(7,13)(11,19)(12,21)(18,20);(6,9,10,17)(7,12,13,21)(8,14,15,16)(11,18,19,20);(6,8,10,15)(7,11,13,19)(9,16,17,14)(12,20,21,18);(4,5)(6,7)(8,11)(9,12)(10,13)(14,18)(15,19)(16,20)(17,21);(1,2)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#69692, #70121, #70347, …
maximal normal subgroup of#69692, #70121, #70347, …
maximal quotient ofC2×#5500, C2×#5513, C2×#5504, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1519);