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G = C37⋊C9order 333 = 32·37

The semidirect product of C37 and C9 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C37⋊C9, C37⋊C3.C3, SmallGroup(333,3)

Series: Derived Chief Lower central Upper central

C1C37 — C37⋊C9
C1C37C37⋊C3 — C37⋊C9
C37 — C37⋊C9
C1

Generators and relations for C37⋊C9
 G = < a,b | a37=b9=1, bab-1=a7 >

37C3
37C9

Character table of C37⋊C9

 class 13A3B9A9B9C9D9E9F37A37B37C37D
 size 137373737373737379999
ρ11111111111111    trivial
ρ2111ζ3ζ32ζ32ζ32ζ3ζ31111    linear of order 3
ρ3111ζ32ζ3ζ3ζ3ζ32ζ321111    linear of order 3
ρ41ζ32ζ3ζ98ζ9ζ94ζ97ζ92ζ951111    linear of order 9
ρ51ζ32ζ3ζ92ζ97ζ9ζ94ζ95ζ981111    linear of order 9
ρ61ζ3ζ32ζ9ζ98ζ95ζ92ζ97ζ941111    linear of order 9
ρ71ζ3ζ32ζ94ζ95ζ92ζ98ζ9ζ971111    linear of order 9
ρ81ζ3ζ32ζ97ζ92ζ98ζ95ζ94ζ91111    linear of order 9
ρ91ζ32ζ3ζ95ζ94ζ97ζ9ζ98ζ921111    linear of order 9
ρ10900000000ζ37323731372937243720371837153714372ζ373537233722371937173713378376375ζ37343733372637163712371037937737ζ3736373037283727372537213711374373    complex faithful
ρ11900000000ζ37343733372637163712371037937737ζ3736373037283727372537213711374373ζ373537233722371937173713378376375ζ37323731372937243720371837153714372    complex faithful
ρ12900000000ζ373537233722371937173713378376375ζ37323731372937243720371837153714372ζ3736373037283727372537213711374373ζ37343733372637163712371037937737    complex faithful
ρ13900000000ζ3736373037283727372537213711374373ζ37343733372637163712371037937737ζ37323731372937243720371837153714372ζ373537233722371937173713378376375    complex faithful

Smallest permutation representation of C37⋊C9
On 37 points: primitive
Generators in S37
(1 2 3 4 5 6 7 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 35 36 37)
(2 17 35 27 10 34 11 13 8)(3 33 32 16 19 30 21 25 15)(4 12 29 5 28 26 31 37 22)(6 7 23 20 9 18 14 24 36)

G:=sub<Sym(37)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22)(6,7,23,20,9,18,14,24,36)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22)(6,7,23,20,9,18,14,24,36) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)], [(2,17,35,27,10,34,11,13,8),(3,33,32,16,19,30,21,25,15),(4,12,29,5,28,26,31,37,22),(6,7,23,20,9,18,14,24,36)])

Matrix representation of C37⋊C9 in GL9(𝔽1999)

010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
154414156987057124671417621
,
100000000
000000010
4971963368754120314951052581463
81881540114615835461519371495
2521744191718951547184218781585702
66814461196999105475761219931455
000001000
124741105316397971818181613831460
117110921262901350861090139795

G:=sub<GL(9,GF(1999))| [0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,544,0,1,0,0,0,0,0,0,1415,0,0,1,0,0,0,0,0,698,0,0,0,1,0,0,0,0,705,0,0,0,0,1,0,0,0,712,0,0,0,0,0,1,0,0,467,0,0,0,0,0,0,1,0,1417,0,0,0,0,0,0,0,1,621],[1,0,497,81,252,668,0,124,1171,0,0,1963,881,1744,1446,0,741,1092,0,0,368,540,1917,1196,0,1053,1262,0,0,754,1146,1895,999,0,1639,90,0,0,1203,1583,1547,1054,0,797,1350,0,0,1495,546,1842,757,1,1818,86,0,0,1052,1519,1878,612,0,1816,1090,0,1,581,371,1585,1993,0,1383,139,0,0,463,495,702,1455,0,1460,795] >;

C37⋊C9 in GAP, Magma, Sage, TeX

C_{37}\rtimes C_9
% in TeX

G:=Group("C37:C9");
// GroupNames label

G:=SmallGroup(333,3);
// by ID

G=gap.SmallGroup(333,3);
# by ID

G:=PCGroup([3,-3,-3,-37,9,1298,707]);
// Polycyclic

G:=Group<a,b|a^37=b^9=1,b*a*b^-1=a^7>;
// generators/relations

Export

Subgroup lattice of C37⋊C9 in TeX
Character table of C37⋊C9 in TeX

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