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G = C6.D18order 216 = 23·33

11st non-split extension by C6 of D18 acting via D18/C18=C2

metabelian, supersoluble, monomial

Aliases: C6.18D18, C18.18D6, C62.12S3, (C3×C9)⋊9D4, (C2×C6)⋊4D9, (C6×C18)⋊5C2, (C2×C18)⋊4S3, C93(C3⋊D4), C33(C9⋊D4), C9⋊Dic34C2, (C3×C6).54D6, C223(C9⋊S3), C3.(C327D4), (C3×C18).22C22, C32.5(C3⋊D4), (C2×C9⋊S3)⋊4C2, C2.5(C2×C9⋊S3), C6.12(C2×C3⋊S3), (C2×C6).6(C3⋊S3), SmallGroup(216,70)

Series: Derived Chief Lower central Upper central

C1C3×C18 — C6.D18
C1C3C32C3×C9C3×C18C2×C9⋊S3 — C6.D18
C3×C9C3×C18 — C6.D18
C1C2C22

Generators and relations for C6.D18
 G = < a,b,c | a6=b18=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 430 in 80 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C22, C22, S3 [×4], C6, C6 [×3], C6 [×4], D4, C9 [×3], C32, Dic3 [×4], D6 [×4], C2×C6, C2×C6 [×3], D9 [×3], C18 [×3], C18 [×3], C3⋊S3, C3×C6, C3×C6, C3⋊D4 [×4], C3×C9, Dic9 [×3], D18 [×3], C2×C18 [×3], C3⋊Dic3, C2×C3⋊S3, C62, C9⋊S3, C3×C18, C3×C18, C9⋊D4 [×3], C327D4, C9⋊Dic3, C2×C9⋊S3, C6×C18, C6.D18
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D9 [×3], C3⋊S3, C3⋊D4 [×4], D18 [×3], C2×C3⋊S3, C9⋊S3, C9⋊D4 [×3], C327D4, C2×C9⋊S3, C6.D18

Smallest permutation representation of C6.D18
On 108 points
Generators in S108
(1 79 26 51 108 55)(2 80 27 52 91 56)(3 81 28 53 92 57)(4 82 29 54 93 58)(5 83 30 37 94 59)(6 84 31 38 95 60)(7 85 32 39 96 61)(8 86 33 40 97 62)(9 87 34 41 98 63)(10 88 35 42 99 64)(11 89 36 43 100 65)(12 90 19 44 101 66)(13 73 20 45 102 67)(14 74 21 46 103 68)(15 75 22 47 104 69)(16 76 23 48 105 70)(17 77 24 49 106 71)(18 78 25 50 107 72)
(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 18 51 50)(2 49 52 17)(3 16 53 48)(4 47 54 15)(5 14 37 46)(6 45 38 13)(7 12 39 44)(8 43 40 11)(9 10 41 42)(19 85 66 96)(20 95 67 84)(21 83 68 94)(22 93 69 82)(23 81 70 92)(24 91 71 80)(25 79 72 108)(26 107 55 78)(27 77 56 106)(28 105 57 76)(29 75 58 104)(30 103 59 74)(31 73 60 102)(32 101 61 90)(33 89 62 100)(34 99 63 88)(35 87 64 98)(36 97 65 86)

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

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

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

C6.D18 is a maximal subgroup of   D18.3D6  Dic3.D18  S3×C9⋊D4  D9×C3⋊D4  C36.70D6  D4×C9⋊S3  C36.27D6
C6.D18 is a maximal quotient of   C6.Dic18  C6.11D36  C36.17D6  C36.18D6  C36.19D6  C36.20D6  C62.127D6

57 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 6A···6L9A···9I18A···18AA
order1222333346···69···918···18
size112542222542···22···22···2

57 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3S3D4D6D6D9C3⋊D4C3⋊D4D18C9⋊D4
kernelC6.D18C9⋊Dic3C2×C9⋊S3C6×C18C2×C18C62C3×C9C18C3×C6C2×C6C9C32C6C3
# reps111131131962918

Matrix representation of C6.D18 in GL4(𝔽37) generated by

363600
1000
00360
00036
,
363600
1000
00213
002426
,
363600
0100
001311
003524
G:=sub<GL(4,GF(37))| [36,1,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[36,1,0,0,36,0,0,0,0,0,2,24,0,0,13,26],[36,0,0,0,36,1,0,0,0,0,13,35,0,0,11,24] >;

C6.D18 in GAP, Magma, Sage, TeX

C_6.D_{18}
% in TeX

G:=Group("C6.D18");
// GroupNames label

G:=SmallGroup(216,70);
// by ID

G=gap.SmallGroup(216,70);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,2115,453,1444,5189]);
// Polycyclic

G:=Group<a,b,c|a^6=b^18=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations

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