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G = C2×C12.58D6order 288 = 25·32

Direct product of C2 and C12.58D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C12.58D6, (C6×C12).25C4, (C3×C6)⋊8M4(2), (C2×C12).400D6, (C2×C62).15C4, C62(C4.Dic3), C62.112(C2×C4), (C22×C12).29S3, C12.61(C2×Dic3), (C2×C12).16Dic3, C3215(C2×M4(2)), (C6×C12).310C22, C12.212(C22×S3), (C3×C12).181C23, C324C831C22, C23.4(C3⋊Dic3), (C22×C6).18Dic3, C6.32(C22×Dic3), (C2×C6×C12).11C2, C33(C2×C4.Dic3), C4.41(C22×C3⋊S3), C4.14(C2×C3⋊Dic3), (C3×C12).126(C2×C4), (C2×C324C8)⋊20C2, (C22×C4).6(C3⋊S3), (C2×C4).6(C3⋊Dic3), (C2×C6).52(C2×Dic3), C2.3(C22×C3⋊Dic3), (C3×C6).121(C22×C4), C22.12(C2×C3⋊Dic3), (C2×C4).83(C2×C3⋊S3), SmallGroup(288,778)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C12.58D6
C1C3C32C3×C6C3×C12C324C8C2×C324C8 — C2×C12.58D6
C32C3×C6 — C2×C12.58D6
C1C2×C4C22×C4

Generators and relations for C2×C12.58D6
 G = < a,b,c,d | a2=b12=c6=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 404 in 204 conjugacy classes, 125 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C32, C12 [×16], C2×C6 [×12], C2×C6 [×8], C2×C8 [×2], M4(2) [×4], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C3⋊C8 [×16], C2×C12 [×24], C22×C6 [×4], C2×M4(2), C3×C12 [×2], C3×C12 [×2], C62, C62 [×2], C62 [×2], C2×C3⋊C8 [×8], C4.Dic3 [×16], C22×C12 [×4], C324C8 [×4], C6×C12 [×2], C6×C12 [×4], C2×C62, C2×C4.Dic3 [×4], C2×C324C8 [×2], C12.58D6 [×4], C2×C6×C12, C2×C12.58D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, Dic3 [×16], D6 [×12], M4(2) [×2], C22×C4, C3⋊S3, C2×Dic3 [×24], C22×S3 [×4], C2×M4(2), C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C4.Dic3 [×8], C22×Dic3 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, C2×C4.Dic3 [×4], C12.58D6 [×2], C22×C3⋊Dic3, C2×C12.58D6

Smallest permutation representation of C2×C12.58D6
On 144 points
Generators in S144
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 105)(26 106)(27 107)(28 108)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 109)(59 110)(60 111)(121 138)(122 139)(123 140)(124 141)(125 142)(126 143)(127 144)(128 133)(129 134)(130 135)(131 136)(132 137)
(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 53 34)(2 54 35)(3 55 36)(4 56 25)(5 57 26)(6 58 27)(7 59 28)(8 60 29)(9 49 30)(10 50 31)(11 51 32)(12 52 33)(13 78 142 19 84 136)(14 79 143 20 73 137)(15 80 144 21 74 138)(16 81 133 22 75 139)(17 82 134 23 76 140)(18 83 135 24 77 141)(37 132 62 43 126 68)(38 121 63 44 127 69)(39 122 64 45 128 70)(40 123 65 46 129 71)(41 124 66 47 130 72)(42 125 67 48 131 61)(85 112 98)(86 113 99)(87 114 100)(88 115 101)(89 116 102)(90 117 103)(91 118 104)(92 119 105)(93 120 106)(94 109 107)(95 110 108)(96 111 97)
(1 22 4 13 7 16 10 19)(2 15 5 18 8 21 11 24)(3 20 6 23 9 14 12 17)(25 84 28 75 31 78 34 81)(26 77 29 80 32 83 35 74)(27 82 30 73 33 76 36 79)(37 101 40 104 43 107 46 98)(38 106 41 97 44 100 47 103)(39 99 42 102 45 105 48 108)(49 143 52 134 55 137 58 140)(50 136 53 139 56 142 59 133)(51 141 54 144 57 135 60 138)(61 95 64 86 67 89 70 92)(62 88 65 91 68 94 71 85)(63 93 66 96 69 87 72 90)(109 123 112 126 115 129 118 132)(110 128 113 131 116 122 119 125)(111 121 114 124 117 127 120 130)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D4E4F6A···6AB8A···8H12A···12AF
order12222233334444446···68···812···12
size11112222221111222···218···182···2

84 irreducible representations

dim111111222222
type+++++-+-
imageC1C2C2C2C4C4S3Dic3D6Dic3M4(2)C4.Dic3
kernelC2×C12.58D6C2×C324C8C12.58D6C2×C6×C12C6×C12C2×C62C22×C12C2×C12C2×C12C22×C6C3×C6C6
# reps124162412124432

Matrix representation of C2×C12.58D6 in GL5(𝔽73)

720000
01000
00100
00010
00001
,
720000
03000
082400
000240
00003
,
720000
01000
0347200
00080
00009
,
460000
027700
044600
000064
000700

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,3,8,0,0,0,0,24,0,0,0,0,0,24,0,0,0,0,0,3],[72,0,0,0,0,0,1,34,0,0,0,0,72,0,0,0,0,0,8,0,0,0,0,0,9],[46,0,0,0,0,0,27,4,0,0,0,7,46,0,0,0,0,0,0,70,0,0,0,64,0] >;

C2×C12.58D6 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{58}D_6
% in TeX

G:=Group("C2xC12.58D6");
// GroupNames label

G:=SmallGroup(288,778);
// by ID

G=gap.SmallGroup(288,778);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

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