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G = C6.Dic6order 144 = 24·32

4th non-split extension by C6 of Dic6 acting via Dic6/C12=C2

metabelian, supersoluble, monomial

Aliases: C6.6Dic6, C62.22C22, (C3×C6).6Q8, C6.14(C4×S3), C328(C4⋊C4), (C2×C12).4S3, (C6×C12).2C2, C3⋊Dic33C4, (C3×C6).33D4, (C2×C6).31D6, C33(Dic3⋊C4), C6.20(C3⋊D4), C2.1(C327D4), C2.1(C324Q8), C2.4(C4×C3⋊S3), (C2×C4).1(C3⋊S3), (C3×C6).25(C2×C4), C22.4(C2×C3⋊S3), (C2×C3⋊Dic3).4C2, SmallGroup(144,93)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C6.Dic6
C1C3C32C3×C6C62C2×C3⋊Dic3 — C6.Dic6
C32C3×C6 — C6.Dic6
C1C22C2×C4

Generators and relations for C6.Dic6
 G = < a,b,c | a6=b12=1, c2=a3b6, ab=ba, cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 202 in 78 conjugacy classes, 41 normal (15 characteristic)
C1, C2 [×3], C3 [×4], C4 [×4], C22, C6 [×12], C2×C4, C2×C4 [×2], C32, Dic3 [×12], C12 [×4], C2×C6 [×4], C4⋊C4, C3×C6 [×3], C2×Dic3 [×8], C2×C12 [×4], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, Dic3⋊C4 [×4], C2×C3⋊Dic3 [×2], C6×C12, C6.Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C2×C3⋊S3, Dic3⋊C4 [×4], C324Q8, C4×C3⋊S3, C327D4, C6.Dic6

Smallest permutation representation of C6.Dic6
Regular action on 144 points
Generators in S144
(1 63 29 86 112 74)(2 64 30 87 113 75)(3 65 31 88 114 76)(4 66 32 89 115 77)(5 67 33 90 116 78)(6 68 34 91 117 79)(7 69 35 92 118 80)(8 70 36 93 119 81)(9 71 25 94 120 82)(10 72 26 95 109 83)(11 61 27 96 110 84)(12 62 28 85 111 73)(13 42 140 99 127 56)(14 43 141 100 128 57)(15 44 142 101 129 58)(16 45 143 102 130 59)(17 46 144 103 131 60)(18 47 133 104 132 49)(19 48 134 105 121 50)(20 37 135 106 122 51)(21 38 136 107 123 52)(22 39 137 108 124 53)(23 40 138 97 125 54)(24 41 139 98 126 55)
(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)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 129 92 38)(2 43 93 122)(3 127 94 48)(4 41 95 132)(5 125 96 46)(6 39 85 130)(7 123 86 44)(8 37 87 128)(9 121 88 42)(10 47 89 126)(11 131 90 40)(12 45 91 124)(13 71 105 114)(14 119 106 64)(15 69 107 112)(16 117 108 62)(17 67 97 110)(18 115 98 72)(19 65 99 120)(20 113 100 70)(21 63 101 118)(22 111 102 68)(23 61 103 116)(24 109 104 66)(25 134 76 56)(26 49 77 139)(27 144 78 54)(28 59 79 137)(29 142 80 52)(30 57 81 135)(31 140 82 50)(32 55 83 133)(33 138 84 60)(34 53 73 143)(35 136 74 58)(36 51 75 141)

G:=sub<Sym(144)| (1,63,29,86,112,74)(2,64,30,87,113,75)(3,65,31,88,114,76)(4,66,32,89,115,77)(5,67,33,90,116,78)(6,68,34,91,117,79)(7,69,35,92,118,80)(8,70,36,93,119,81)(9,71,25,94,120,82)(10,72,26,95,109,83)(11,61,27,96,110,84)(12,62,28,85,111,73)(13,42,140,99,127,56)(14,43,141,100,128,57)(15,44,142,101,129,58)(16,45,143,102,130,59)(17,46,144,103,131,60)(18,47,133,104,132,49)(19,48,134,105,121,50)(20,37,135,106,122,51)(21,38,136,107,123,52)(22,39,137,108,124,53)(23,40,138,97,125,54)(24,41,139,98,126,55), (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)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,129,92,38)(2,43,93,122)(3,127,94,48)(4,41,95,132)(5,125,96,46)(6,39,85,130)(7,123,86,44)(8,37,87,128)(9,121,88,42)(10,47,89,126)(11,131,90,40)(12,45,91,124)(13,71,105,114)(14,119,106,64)(15,69,107,112)(16,117,108,62)(17,67,97,110)(18,115,98,72)(19,65,99,120)(20,113,100,70)(21,63,101,118)(22,111,102,68)(23,61,103,116)(24,109,104,66)(25,134,76,56)(26,49,77,139)(27,144,78,54)(28,59,79,137)(29,142,80,52)(30,57,81,135)(31,140,82,50)(32,55,83,133)(33,138,84,60)(34,53,73,143)(35,136,74,58)(36,51,75,141)>;

G:=Group( (1,63,29,86,112,74)(2,64,30,87,113,75)(3,65,31,88,114,76)(4,66,32,89,115,77)(5,67,33,90,116,78)(6,68,34,91,117,79)(7,69,35,92,118,80)(8,70,36,93,119,81)(9,71,25,94,120,82)(10,72,26,95,109,83)(11,61,27,96,110,84)(12,62,28,85,111,73)(13,42,140,99,127,56)(14,43,141,100,128,57)(15,44,142,101,129,58)(16,45,143,102,130,59)(17,46,144,103,131,60)(18,47,133,104,132,49)(19,48,134,105,121,50)(20,37,135,106,122,51)(21,38,136,107,123,52)(22,39,137,108,124,53)(23,40,138,97,125,54)(24,41,139,98,126,55), (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)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,129,92,38)(2,43,93,122)(3,127,94,48)(4,41,95,132)(5,125,96,46)(6,39,85,130)(7,123,86,44)(8,37,87,128)(9,121,88,42)(10,47,89,126)(11,131,90,40)(12,45,91,124)(13,71,105,114)(14,119,106,64)(15,69,107,112)(16,117,108,62)(17,67,97,110)(18,115,98,72)(19,65,99,120)(20,113,100,70)(21,63,101,118)(22,111,102,68)(23,61,103,116)(24,109,104,66)(25,134,76,56)(26,49,77,139)(27,144,78,54)(28,59,79,137)(29,142,80,52)(30,57,81,135)(31,140,82,50)(32,55,83,133)(33,138,84,60)(34,53,73,143)(35,136,74,58)(36,51,75,141) );

G=PermutationGroup([(1,63,29,86,112,74),(2,64,30,87,113,75),(3,65,31,88,114,76),(4,66,32,89,115,77),(5,67,33,90,116,78),(6,68,34,91,117,79),(7,69,35,92,118,80),(8,70,36,93,119,81),(9,71,25,94,120,82),(10,72,26,95,109,83),(11,61,27,96,110,84),(12,62,28,85,111,73),(13,42,140,99,127,56),(14,43,141,100,128,57),(15,44,142,101,129,58),(16,45,143,102,130,59),(17,46,144,103,131,60),(18,47,133,104,132,49),(19,48,134,105,121,50),(20,37,135,106,122,51),(21,38,136,107,123,52),(22,39,137,108,124,53),(23,40,138,97,125,54),(24,41,139,98,126,55)], [(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),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,129,92,38),(2,43,93,122),(3,127,94,48),(4,41,95,132),(5,125,96,46),(6,39,85,130),(7,123,86,44),(8,37,87,128),(9,121,88,42),(10,47,89,126),(11,131,90,40),(12,45,91,124),(13,71,105,114),(14,119,106,64),(15,69,107,112),(16,117,108,62),(17,67,97,110),(18,115,98,72),(19,65,99,120),(20,113,100,70),(21,63,101,118),(22,111,102,68),(23,61,103,116),(24,109,104,66),(25,134,76,56),(26,49,77,139),(27,144,78,54),(28,59,79,137),(29,142,80,52),(30,57,81,135),(31,140,82,50),(32,55,83,133),(33,138,84,60),(34,53,73,143),(35,136,74,58),(36,51,75,141)])

C6.Dic6 is a maximal subgroup of
Dic35Dic6  C62.9C23  C62.16C23  C62.17C23  D6⋊Dic6  C62.28C23  C62.29C23  C62.37C23  S3×Dic3⋊C4  C62.47C23  C62.49C23  C62.54C23  C62.55C23  D61Dic6  D64Dic6  C62.75C23  C4×C324Q8  C12.25Dic6  C12216C2  C1222C2  C62.221C23  C626Q8  C62.223C23  C62.225C23  C62.227C23  C62.228C23  C62.231C23  C122Dic6  C62.233C23  C62.234C23  C4⋊C4×C3⋊S3  C62.238C23  C62.240C23  C62.242C23  C6210Q8  C4×C327D4  C62.129D4  C62.72D4  C6214D4  C62.259C23  C62.261C23  C62.19D6  C6.Dic18  C62.81D6  C62.82D6  C62.146D6
C6.Dic6 is a maximal quotient of
C12.9Dic6  C12.10Dic6  C12.30Dic6  C62.8Q8  C62.15Q8  C6.Dic18  C62.29D6  C62.81D6  C62.82D6  C62.146D6

42 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L12A···12P
order122233334444446···612···12
size1111222222181818182···22···2

42 irreducible representations

dim11112222222
type+++++-+-
imageC1C2C2C4S3D4Q8D6Dic6C4×S3C3⋊D4
kernelC6.Dic6C2×C3⋊Dic3C6×C12C3⋊Dic3C2×C12C3×C6C3×C6C2×C6C6C6C6
# reps12144114888

Matrix representation of C6.Dic6 in GL4(𝔽13) generated by

10000
0400
0001
00121
,
8000
0800
0022
00114
,
0100
1000
0080
0085
G:=sub<GL(4,GF(13))| [10,0,0,0,0,4,0,0,0,0,0,12,0,0,1,1],[8,0,0,0,0,8,0,0,0,0,2,11,0,0,2,4],[0,1,0,0,1,0,0,0,0,0,8,8,0,0,0,5] >;

C6.Dic6 in GAP, Magma, Sage, TeX

C_6.{\rm Dic}_6
% in TeX

G:=Group("C6.Dic6");
// GroupNames label

G:=SmallGroup(144,93);
// by ID

G=gap.SmallGroup(144,93);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^6=b^12=1,c^2=a^3*b^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations

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