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## G = C62⋊19D4order 288 = 25·32

### 3rd semidirect product of C62 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊19D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C12⋊S3 — C62⋊19D4
 Lower central C32 — C62 — C62⋊19D4
 Upper central C1 — C22 — C22×C4

Generators and relations for C6219D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, ac=ca, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1196 in 282 conjugacy classes, 89 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3 [×8], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×8], C12 [×8], C12 [×4], D6 [×24], C2×C6 [×12], C2×C6 [×8], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×2], D12 [×8], C2×Dic3 [×8], C3⋊D4 [×16], C2×C12 [×8], C2×C12 [×8], C22×S3 [×8], C22×C6 [×4], C4⋊D4, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, C2×C3⋊S3 [×6], C62, C62 [×2], C62 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C2×D12 [×4], C2×C3⋊D4 [×8], C22×C12 [×4], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12 [×2], C6×C12 [×2], C22×C3⋊S3 [×2], C2×C62, C127D4 [×4], C12⋊Dic3, C6.11D12 [×2], C2×C12⋊S3, C2×C327D4 [×2], C2×C6×C12, C6219D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, D12 [×8], C3⋊D4 [×8], C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], C2×D12 [×4], C4○D12 [×4], C2×C3⋊D4 [×4], C12⋊S3 [×2], C327D4 [×2], C22×C3⋊S3, C127D4 [×4], C2×C12⋊S3, C12.59D6, C2×C327D4, C6219D4

Smallest permutation representation of C6219D4
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 75 35 18 25 112)(2 76 36 13 26 113)(3 77 31 14 27 114)(4 78 32 15 28 109)(5 73 33 16 29 110)(6 74 34 17 30 111)(7 106 87 102 47 81)(8 107 88 97 48 82)(9 108 89 98 43 83)(10 103 90 99 44 84)(11 104 85 100 45 79)(12 105 86 101 46 80)(19 69 65 52 117 56)(20 70 66 53 118 57)(21 71 61 54 119 58)(22 72 62 49 120 59)(23 67 63 50 115 60)(24 68 64 51 116 55)(37 128 91 141 137 124)(38 129 92 142 138 125)(39 130 93 143 133 126)(40 131 94 144 134 121)(41 132 95 139 135 122)(42 127 96 140 136 123)
(1 132 60 82)(2 127 55 83)(3 128 56 84)(4 129 57 79)(5 130 58 80)(6 131 59 81)(7 74 94 22)(8 75 95 23)(9 76 96 24)(10 77 91 19)(11 78 92 20)(12 73 93 21)(13 136 64 89)(14 137 65 90)(15 138 66 85)(16 133 61 86)(17 134 62 87)(18 135 63 88)(25 122 50 97)(26 123 51 98)(27 124 52 99)(28 125 53 100)(29 126 54 101)(30 121 49 102)(31 141 69 103)(32 142 70 104)(33 143 71 105)(34 144 72 106)(35 139 67 107)(36 140 68 108)(37 117 44 114)(38 118 45 109)(39 119 46 110)(40 120 47 111)(41 115 48 112)(42 116 43 113)
(1 82)(2 87)(3 80)(4 85)(5 84)(6 89)(7 36)(8 112)(9 34)(10 110)(11 32)(12 114)(13 81)(14 86)(15 79)(16 90)(17 83)(18 88)(19 39)(20 142)(21 37)(22 140)(23 41)(24 144)(25 107)(26 47)(27 105)(28 45)(29 103)(30 43)(31 101)(33 99)(35 97)(38 53)(40 51)(42 49)(44 73)(46 77)(48 75)(50 139)(52 143)(54 141)(55 134)(56 130)(57 138)(58 128)(59 136)(60 132)(61 137)(62 127)(63 135)(64 131)(65 133)(66 129)(67 122)(68 94)(69 126)(70 92)(71 124)(72 96)(74 108)(76 106)(78 104)(91 119)(93 117)(95 115)(98 111)(100 109)(102 113)(116 121)(118 125)(120 123)```

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6AB 12A ··· 12AF order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 36 36 2 2 2 2 2 2 2 2 36 36 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 D12 C4○D12 kernel C62⋊19D4 C12⋊Dic3 C6.11D12 C2×C12⋊S3 C2×C32⋊7D4 C2×C6×C12 C22×C12 C3×C12 C62 C2×C12 C22×C6 C3×C6 C12 C2×C6 C6 # reps 1 1 2 1 2 1 4 2 2 8 4 2 16 16 16

Matrix representation of C6219D4 in GL4(𝔽13) generated by

 0 12 0 0 1 1 0 0 0 0 2 4 0 0 9 11
,
 12 12 0 0 1 0 0 0 0 0 1 1 0 0 12 0
,
 3 6 0 0 7 10 0 0 0 0 12 0 0 0 0 12
,
 10 7 0 0 10 3 0 0 0 0 12 0 0 0 1 1
`G:=sub<GL(4,GF(13))| [0,1,0,0,12,1,0,0,0,0,2,9,0,0,4,11],[12,1,0,0,12,0,0,0,0,0,1,12,0,0,1,0],[3,7,0,0,6,10,0,0,0,0,12,0,0,0,0,12],[10,10,0,0,7,3,0,0,0,0,12,1,0,0,0,1] >;`

C6219D4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_{19}D_4`
`% in TeX`

`G:=Group("C6^2:19D4");`
`// GroupNames label`

`G:=SmallGroup(288,787);`
`// by ID`

`G=gap.SmallGroup(288,787);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,2693,9414]);`
`// Polycyclic`

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

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