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G = C12.31D12order 288 = 25·32

31st non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.31D12, C62.241C23, C6.47(S3×Q8), (C3×C12).94D4, (C2×C12).35D6, C6.55(C2×D12), C33(C4.D12), C12⋊Dic39C2, (C6×C12).18C22, C4.13(C12⋊S3), C3221(C22⋊Q8), C6.11D12.3C2, C6.102(D42S3), C2.13(C12.D6), (C3×C4⋊C4)⋊8S3, C4⋊C45(C3⋊S3), (C2×C3⋊S3)⋊8Q8, C2.7(Q8×C3⋊S3), (C3×C6).74(C2×Q8), (C32×C4⋊C4)⋊17C2, (C3×C6).195(C2×D4), C2.10(C2×C12⋊S3), (C2×C324Q8)⋊13C2, (C3×C6).149(C4○D4), (C2×C6).258(C22×S3), C22.52(C22×C3⋊S3), (C22×C3⋊S3).89C22, (C2×C3⋊Dic3).87C22, (C2×C4×C3⋊S3).6C2, (C2×C4).14(C2×C3⋊S3), SmallGroup(288,754)

Series: Derived Chief Lower central Upper central

C1C62 — C12.31D12
C1C3C32C3×C6C62C22×C3⋊S3C2×C4×C3⋊S3 — C12.31D12
C32C62 — C12.31D12
C1C22C4⋊C4

Generators and relations for C12.31D12
 G = < a,b,c | a12=b12=1, c2=a6, bab-1=a7, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 860 in 222 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊Q8, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, C4.D12, C12⋊Dic3, C6.11D12, C32×C4⋊C4, C2×C324Q8, C2×C4×C3⋊S3, C12.31D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊S3, D12, C22×S3, C22⋊Q8, C2×C3⋊S3, C2×D12, D42S3, S3×Q8, C12⋊S3, C22×C3⋊S3, C4.D12, C2×C12⋊S3, C12.D6, Q8×C3⋊S3, C12.31D12

Smallest permutation representation of C12.31D12
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 86 82 14 123 108 37 141 54 35 62 120)(2 93 83 21 124 103 38 136 55 30 63 115)(3 88 84 16 125 98 39 143 56 25 64 110)(4 95 73 23 126 105 40 138 57 32 65 117)(5 90 74 18 127 100 41 133 58 27 66 112)(6 85 75 13 128 107 42 140 59 34 67 119)(7 92 76 20 129 102 43 135 60 29 68 114)(8 87 77 15 130 97 44 142 49 36 69 109)(9 94 78 22 131 104 45 137 50 31 70 116)(10 89 79 17 132 99 46 144 51 26 71 111)(11 96 80 24 121 106 47 139 52 33 72 118)(12 91 81 19 122 101 48 134 53 28 61 113)
(1 114 7 120)(2 113 8 119)(3 112 9 118)(4 111 10 117)(5 110 11 116)(6 109 12 115)(13 55 19 49)(14 54 20 60)(15 53 21 59)(16 52 22 58)(17 51 23 57)(18 50 24 56)(25 80 31 74)(26 79 32 73)(27 78 33 84)(28 77 34 83)(29 76 35 82)(30 75 36 81)(37 102 43 108)(38 101 44 107)(39 100 45 106)(40 99 46 105)(41 98 47 104)(42 97 48 103)(61 93 67 87)(62 92 68 86)(63 91 69 85)(64 90 70 96)(65 89 71 95)(66 88 72 94)(121 137 127 143)(122 136 128 142)(123 135 129 141)(124 134 130 140)(125 133 131 139)(126 144 132 138)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D4E4F4G4H6A···6L12A···12X
order1222223333444444446···612···12
size1111181822222244181836362···24···4

54 irreducible representations

dim11111122222244
type++++++++-++--
imageC1C2C2C2C2C2S3D4Q8D6C4○D4D12D42S3S3×Q8
kernelC12.31D12C12⋊Dic3C6.11D12C32×C4⋊C4C2×C324Q8C2×C4×C3⋊S3C3×C4⋊C4C3×C12C2×C3⋊S3C2×C12C3×C6C12C6C6
# reps1221114221221644

Matrix representation of C12.31D12 in GL6(𝔽13)

0120000
110000
001000
000100
000050
000008
,
360000
7100000
00121200
001000
000001
000010
,
360000
3100000
0012000
001100
000001
0000120

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

C12.31D12 in GAP, Magma, Sage, TeX

C_{12}._{31}D_{12}
% in TeX

G:=Group("C12.31D12");
// GroupNames label

G:=SmallGroup(288,754);
// by ID

G=gap.SmallGroup(288,754);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,142,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^6,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations

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