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G = C12⋊Dic3order 144 = 24·32

1st semidirect product of C12 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: C121Dic3, C6.10D12, C6.7Dic6, C62.23C22, (C3×C12)⋊3C4, C4⋊(C3⋊Dic3), (C3×C6).7Q8, C329(C4⋊C4), (C6×C12).4C2, (C2×C12).8S3, (C3×C6).25D4, (C2×C6).32D6, C32(C4⋊Dic3), C6.15(C2×Dic3), C2.1(C12⋊S3), C2.2(C324Q8), (C2×C4).3(C3⋊S3), (C3×C6).33(C2×C4), C22.5(C2×C3⋊S3), C2.4(C2×C3⋊Dic3), (C2×C3⋊Dic3).5C2, SmallGroup(144,94)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12⋊Dic3
C1C3C32C3×C6C62C2×C3⋊Dic3 — C12⋊Dic3
C32C3×C6 — C12⋊Dic3
C1C22C2×C4

Generators and relations for C12⋊Dic3
 G = < a,b,c | a12=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 202 in 78 conjugacy classes, 51 normal (13 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×8], C2×C6 [×4], C4⋊C4, C3×C6 [×3], C2×Dic3 [×8], C2×C12 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C12⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4⋊Dic3 [×4], C324Q8, C12⋊S3, C2×C3⋊Dic3, C12⋊Dic3

Smallest permutation representation of C12⋊Dic3
Regular action on 144 points
Generators in S144
(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)
(1 126 46 64 29 135)(2 127 47 65 30 136)(3 128 48 66 31 137)(4 129 37 67 32 138)(5 130 38 68 33 139)(6 131 39 69 34 140)(7 132 40 70 35 141)(8 121 41 71 36 142)(9 122 42 72 25 143)(10 123 43 61 26 144)(11 124 44 62 27 133)(12 125 45 63 28 134)(13 95 51 81 104 117)(14 96 52 82 105 118)(15 85 53 83 106 119)(16 86 54 84 107 120)(17 87 55 73 108 109)(18 88 56 74 97 110)(19 89 57 75 98 111)(20 90 58 76 99 112)(21 91 59 77 100 113)(22 92 60 78 101 114)(23 93 49 79 102 115)(24 94 50 80 103 116)
(1 108 64 87)(2 107 65 86)(3 106 66 85)(4 105 67 96)(5 104 68 95)(6 103 69 94)(7 102 70 93)(8 101 71 92)(9 100 72 91)(10 99 61 90)(11 98 62 89)(12 97 63 88)(13 130 81 33)(14 129 82 32)(15 128 83 31)(16 127 84 30)(17 126 73 29)(18 125 74 28)(19 124 75 27)(20 123 76 26)(21 122 77 25)(22 121 78 36)(23 132 79 35)(24 131 80 34)(37 52 138 118)(38 51 139 117)(39 50 140 116)(40 49 141 115)(41 60 142 114)(42 59 143 113)(43 58 144 112)(44 57 133 111)(45 56 134 110)(46 55 135 109)(47 54 136 120)(48 53 137 119)

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

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

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

C12⋊Dic3 is a maximal subgroup of
C6.16D24  C6.Dic12  C12.Dic6  C6.18D24  C12.9Dic6  C12.10Dic6  C6.4Dic12  C242Dic3  C241Dic3  C62.84D4  C62.116D4  C62.117D4  C62.11C23  Dic3×Dic6  Dic3.Dic6  D66Dic6  C62.31C23  C62.39C23  S3×C4⋊Dic3  D6.9D12  Dic3×D12  D62Dic6  D62D12  C123Dic6  C4×C324Q8  C126Dic6  C12.25Dic6  C4×C12⋊S3  C626Q8  C62.223C23  C62.227C23  C62.69D4  C122Dic6  C62.233C23  C62.234C23  C4⋊C4×C3⋊S3  C62.236C23  C12.31D12  C62.242C23  C6210Q8  C62.247C23  C6219D4  D4×C3⋊Dic3  C62.256C23  Q8×C3⋊Dic3  C62.261C23  C62.20D6  C36⋊Dic3  C62.80D6  C62.147D6
C12⋊Dic3 is a maximal quotient of
C12.57D12  C242Dic3  C241Dic3  C12.59D12  C62.15Q8  C36⋊Dic3  C62.30D6  C62.80D6  C62.147D6

42 conjugacy classes

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

42 irreducible representations

dim11112222222
type+++++--+-+
imageC1C2C2C4S3D4Q8Dic3D6Dic6D12
kernelC12⋊Dic3C2×C3⋊Dic3C6×C12C3×C12C2×C12C3×C6C3×C6C12C2×C6C6C6
# reps12144118488

Matrix representation of C12⋊Dic3 in GL6(𝔽13)

10100000
370000
000100
00121200
0000112
000010
,
100000
010000
0012000
0001200
0000121
0000120
,
1120000
0120000
005000
008800
0000106
000033

G:=sub<GL(6,GF(13))| [10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,5,8,0,0,0,0,0,8,0,0,0,0,0,0,10,3,0,0,0,0,6,3] >;

C12⋊Dic3 in GAP, Magma, Sage, TeX

C_{12}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C12:Dic3");
// GroupNames label

G:=SmallGroup(144,94);
// by ID

G=gap.SmallGroup(144,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,55,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^12=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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