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G = C2×C12.58D6order 288 = 25·32

Direct product of C2 and C12.58D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12.58D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C2×C32⋊4C8 — C2×C12.58D6
 Lower central C32 — C3×C6 — C2×C12.58D6
 Upper central C1 — C2×C4 — C22×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 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6AB 8A ··· 8H 12A ··· 12AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 ··· 2 18 ··· 18 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 kernel C2×C12.58D6 C2×C32⋊4C8 C12.58D6 C2×C6×C12 C6×C12 C2×C62 C22×C12 C2×C12 C2×C12 C22×C6 C3×C6 C6 # reps 1 2 4 1 6 2 4 12 12 4 4 32

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

 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 0 0 0 0 8 24 0 0 0 0 0 24 0 0 0 0 0 3
,
 72 0 0 0 0 0 1 0 0 0 0 34 72 0 0 0 0 0 8 0 0 0 0 0 9
,
 46 0 0 0 0 0 27 7 0 0 0 4 46 0 0 0 0 0 0 64 0 0 0 70 0

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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