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G = C9⋊Dic3order 108 = 22·33

The semidirect product of C9 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, A-group

Aliases: C9⋊Dic3, C3⋊Dic9, C6.3D9, C18.3S3, C32.3Dic3, (C3×C9)⋊3C4, C2.(C9⋊S3), (C3×C6).6S3, C6.1(C3⋊S3), (C3×C18).3C2, C3.(C3⋊Dic3), SmallGroup(108,10)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C9⋊Dic3
C1C3C32C3×C9C3×C18 — C9⋊Dic3
C3×C9 — C9⋊Dic3
C1C2

Generators and relations for C9⋊Dic3
 G = < a,b,c | a9=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

27C4
9Dic3
9Dic3
9Dic3
9Dic3
3Dic9
3C3⋊Dic3
3Dic9
3Dic9

Character table of C9⋊Dic3

 class 123A3B3C3D4A4B6A6B6C6D9A9B9C9D9E9F9G9H9I18A18B18C18D18E18F18G18H18I
 size 11222227272222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ2111111-1-11111111111111111111111    linear of order 2
ρ31-11111-ii-1-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-11111i-i-1-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 4
ρ522-1-1-1200-12-1-12-1-122-1-1-1-1-1222-1-1-1-1-1    orthogonal lifted from S3
ρ6222222002222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ722-1-1-1200-12-1-1-12-1-1-122-1-1-1-1-1-1222-1-1    orthogonal lifted from S3
ρ8222-1-1-100-1-12-1ζ9594ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ989ζ9792ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ922-1-1-1200-12-1-1-1-12-1-1-1-1222-1-1-1-1-1-122    orthogonal lifted from S3
ρ1022-1-12-1002-1-1-1ζ9792ζ9792ζ989ζ9594ζ989ζ9594ζ989ζ9792ζ9594ζ9594ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ989ζ9792    orthogonal lifted from D9
ρ11222-1-1-100-1-12-1ζ989ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9792ζ9594ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ12222-1-1-100-1-12-1ζ9792ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ9594ζ989ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1322-1-12-1002-1-1-1ζ989ζ989ζ9594ζ9792ζ9594ζ9792ζ9594ζ989ζ9792ζ9792ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ9594ζ989    orthogonal lifted from D9
ρ1422-12-1-100-1-1-12ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ9792ζ989ζ9792ζ9792ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9594ζ989    orthogonal lifted from D9
ρ1522-12-1-100-1-1-12ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ989ζ9594ζ989ζ989ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ9792ζ9594    orthogonal lifted from D9
ρ1622-1-12-1002-1-1-1ζ9594ζ9594ζ9792ζ989ζ9792ζ989ζ9792ζ9594ζ989ζ989ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9792ζ9594    orthogonal lifted from D9
ρ1722-12-1-100-1-1-12ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9594ζ9792ζ9594ζ9594ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ989ζ9792    orthogonal lifted from D9
ρ182-2-1-1-12001-2112-1-122-1-1-1-11-2-2-211111    symplectic lifted from Dic3, Schur index 2
ρ192-2-1-1-12001-211-12-1-1-122-1-11111-2-2-211    symplectic lifted from Dic3, Schur index 2
ρ202-22-1-1-10011-21ζ9594ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ989ζ9792979295949899792959498997929594989    symplectic lifted from Dic9, Schur index 2
ρ212-2-1-1-12001-211-1-12-1-1-1-122-2111111-2-2    symplectic lifted from Dic3, Schur index 2
ρ222-2-12-1-100111-2ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9594ζ9792ζ9594959495949899792979295949899899792    symplectic lifted from Dic9, Schur index 2
ρ232-2-12-1-100111-2ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ9792ζ989ζ9792979297929594989989979295949594989    symplectic lifted from Dic9, Schur index 2
ρ242-2-1-12-100-2111ζ989ζ989ζ9594ζ9792ζ9594ζ9792ζ9594ζ989ζ9792979298997929594979295949899594989    symplectic lifted from Dic9, Schur index 2
ρ252-2-1-12-100-2111ζ9594ζ9594ζ9792ζ989ζ9792ζ989ζ9792ζ9594ζ989989959498997929899792959497929594    symplectic lifted from Dic9, Schur index 2
ρ262-22-1-1-10011-21ζ9792ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ9594ζ989989979295949899792959498997929594    symplectic lifted from Dic9, Schur index 2
ρ272-2222200-2-2-2-2-1-1-1-1-1-1-1-1-1111111111    symplectic lifted from Dic3, Schur index 2
ρ282-2-12-1-100111-2ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ989ζ9594ζ989989989979295949594989979297929594    symplectic lifted from Dic9, Schur index 2
ρ292-2-1-12-100-2111ζ9792ζ9792ζ989ζ9594ζ989ζ9594ζ989ζ9792ζ9594959497929594989959498997929899792    symplectic lifted from Dic9, Schur index 2
ρ302-22-1-1-10011-21ζ989ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9792ζ9594959498997929594989979295949899792    symplectic lifted from Dic9, Schur index 2

Smallest permutation representation of C9⋊Dic3
Regular action on 108 points
Generators in S108
(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 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 48 35 59 37 65)(2 49 36 60 38 66)(3 50 28 61 39 67)(4 51 29 62 40 68)(5 52 30 63 41 69)(6 53 31 55 42 70)(7 54 32 56 43 71)(8 46 33 57 44 72)(9 47 34 58 45 64)(10 80 25 98 107 84)(11 81 26 99 108 85)(12 73 27 91 100 86)(13 74 19 92 101 87)(14 75 20 93 102 88)(15 76 21 94 103 89)(16 77 22 95 104 90)(17 78 23 96 105 82)(18 79 24 97 106 83)
(1 23 59 82)(2 22 60 90)(3 21 61 89)(4 20 62 88)(5 19 63 87)(6 27 55 86)(7 26 56 85)(8 25 57 84)(9 24 58 83)(10 72 98 33)(11 71 99 32)(12 70 91 31)(13 69 92 30)(14 68 93 29)(15 67 94 28)(16 66 95 36)(17 65 96 35)(18 64 97 34)(37 105 48 78)(38 104 49 77)(39 103 50 76)(40 102 51 75)(41 101 52 74)(42 100 53 73)(43 108 54 81)(44 107 46 80)(45 106 47 79)

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

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

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

C9⋊Dic3 is a maximal subgroup of
C9⋊Dic6  Dic3×D9  S3×Dic9  D6⋊D9  C12.D9  C4×C9⋊S3  C6.D18  C32⋊Dic9  He3.Dic3  He3.2Dic3  C9⋊Dic9  C27⋊Dic3  C33.Dic3  He3.4Dic3  C325Dic9  C18.5S4  A4⋊Dic9  C32.3CSU2(𝔽3)  C62.10Dic3
C9⋊Dic3 is a maximal quotient of
C36.S3  C9⋊Dic9  C322Dic9  C27⋊Dic3  C325Dic9  A4⋊Dic9  C62.10Dic3

Matrix representation of C9⋊Dic3 in GL4(𝔽37) generated by

171100
26600
00266
003120
,
1100
36000
0001
003636
,
71400
73000
0001
0010
G:=sub<GL(4,GF(37))| [17,26,0,0,11,6,0,0,0,0,26,31,0,0,6,20],[1,36,0,0,1,0,0,0,0,0,0,36,0,0,1,36],[7,7,0,0,14,30,0,0,0,0,0,1,0,0,1,0] >;

C9⋊Dic3 in GAP, Magma, Sage, TeX

C_9\rtimes {\rm Dic}_3
% in TeX

G:=Group("C9:Dic3");
// GroupNames label

G:=SmallGroup(108,10);
// by ID

G=gap.SmallGroup(108,10);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,10,662,282,483,1804]);
// Polycyclic

G:=Group<a,b,c|a^9=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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Subgroup lattice of C9⋊Dic3 in TeX
Character table of C9⋊Dic3 in TeX

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