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## G = C62⋊14D4order 288 = 25·32

### 11st semidirect product of C62 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊14D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C32⋊7D4 — C62⋊14D4
 Lower central C32 — C62 — C62⋊14D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C6214D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1068 in 282 conjugacy classes, 79 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×5], C22, C22 [×2], C22 [×8], S3 [×4], C6 [×12], C6 [×12], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C32, Dic3 [×16], C12 [×4], D6 [×12], C2×C6 [×12], C2×C6 [×20], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C3⋊S3, C3×C6 [×3], C3×C6 [×3], C2×Dic3 [×20], C3⋊D4 [×16], C2×C12 [×4], C3×D4 [×8], C22×S3 [×4], C22×C6 [×8], C4⋊D4, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3 [×3], C62, C62 [×2], C62 [×5], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], C22×Dic3 [×4], C2×C3⋊D4 [×8], C6×D4 [×4], C2×C3⋊Dic3 [×3], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12, D4×C32 [×2], C22×C3⋊S3, C2×C62 [×2], C23.14D6 [×4], C6.Dic6, C6.11D12, C625C4, C22×C3⋊Dic3, C2×C327D4 [×2], D4×C3×C6, C6214D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, C23.14D6 [×4], D4×C3⋊S3, C12.D6, C2×C327D4, C6214D4

Smallest permutation representation of C6214D4
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 20 119 45 16 121)(2 21 120 46 17 122)(3 22 115 47 18 123)(4 23 116 48 13 124)(5 24 117 43 14 125)(6 19 118 44 15 126)(7 85 74 49 114 81)(8 86 75 50 109 82)(9 87 76 51 110 83)(10 88 77 52 111 84)(11 89 78 53 112 79)(12 90 73 54 113 80)(25 135 61 103 143 42)(26 136 62 104 144 37)(27 137 63 105 139 38)(28 138 64 106 140 39)(29 133 65 107 141 40)(30 134 66 108 142 41)(31 71 102 94 57 130)(32 72 97 95 58 131)(33 67 98 96 59 132)(34 68 99 91 60 127)(35 69 100 92 55 128)(36 70 101 93 56 129)
(1 38 101 52)(2 37 102 51)(3 42 97 50)(4 41 98 49)(5 40 99 54)(6 39 100 53)(7 48 66 132)(8 47 61 131)(9 46 62 130)(10 45 63 129)(11 44 64 128)(12 43 65 127)(13 134 33 81)(14 133 34 80)(15 138 35 79)(16 137 36 84)(17 136 31 83)(18 135 32 82)(19 140 92 78)(20 139 93 77)(21 144 94 76)(22 143 95 75)(23 142 96 74)(24 141 91 73)(25 72 109 123)(26 71 110 122)(27 70 111 121)(28 69 112 126)(29 68 113 125)(30 67 114 124)(55 89 118 106)(56 88 119 105)(57 87 120 104)(58 86 115 103)(59 85 116 108)(60 90 117 107)
(2 44)(3 5)(4 48)(6 46)(7 41)(8 65)(9 39)(10 63)(11 37)(12 61)(13 124)(14 115)(15 122)(16 119)(17 126)(18 117)(19 120)(20 121)(21 118)(22 125)(23 116)(24 123)(25 73)(26 79)(27 77)(28 83)(29 75)(30 81)(31 69)(32 60)(33 67)(34 58)(35 71)(36 56)(38 52)(40 50)(42 54)(43 47)(49 66)(51 64)(53 62)(55 94)(57 92)(59 96)(68 95)(70 93)(72 91)(74 108)(76 106)(78 104)(80 103)(82 107)(84 105)(85 142)(86 133)(87 140)(88 137)(89 144)(90 135)(97 99)(98 132)(100 130)(102 128)(109 141)(110 138)(111 139)(112 136)(113 143)(114 134)(127 131)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 6M ··· 6AB 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 4 36 2 2 2 2 4 18 18 18 18 36 2 ··· 2 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C62⋊14D4 C6.Dic6 C6.11D12 C62⋊5C4 C22×C3⋊Dic3 C2×C32⋊7D4 D4×C3×C6 C6×D4 C3⋊Dic3 C62 C2×C12 C22×C6 C3×C6 C2×C6 C6 C6 # reps 1 1 1 1 1 2 1 4 2 2 4 8 2 16 4 4

Matrix representation of C6214D4 in GL6(𝔽13)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 4 11 0 0 0 0 2 9 0 0 0 0 0 0 9 2 0 0 0 0 11 4
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C6214D4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_{14}D_4`
`% in TeX`

`G:=Group("C6^2:14D4");`
`// GroupNames label`

`G:=SmallGroup(288,796);`
`// by ID`

`G=gap.SmallGroup(288,796);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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