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## G = C62.84D4order 288 = 25·32

### 9th non-split extension by C62 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.84D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C12⋊S3 — C62.84D4
 Lower central C32 — C3×C6 — C3×C12 — C62.84D4
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C62.84D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 836 in 150 conjugacy classes, 59 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4, C22, C22 [×4], S3 [×8], C6 [×12], C8, C2×C4, C2×C4, D4 [×3], C23, C32, Dic3 [×4], C12 [×8], D6 [×16], C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C3⋊S3 [×2], C3×C6 [×3], C24 [×4], D12 [×12], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], D4⋊C4, C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×4], C62, C4⋊Dic3 [×4], C2×C24 [×4], C2×D12 [×4], C3×C24, C12⋊S3 [×2], C12⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2.D24 [×4], C12⋊Dic3, C6×C24, C2×C12⋊S3, C62.84D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, D8, SD16, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], D4⋊C4, C2×C3⋊S3, C24⋊C2 [×4], D24 [×4], D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C2.D24 [×4], C242S3, C325D8, C6.11D12, C62.84D4

Smallest permutation representation of C62.84D4
On 144 points
Generators in S144
```(1 50 141 100 69 27)(2 51 142 101 70 28)(3 52 143 102 71 29)(4 53 144 103 72 30)(5 54 137 104 65 31)(6 55 138 97 66 32)(7 56 139 98 67 25)(8 49 140 99 68 26)(9 21 45 35 74 135)(10 22 46 36 75 136)(11 23 47 37 76 129)(12 24 48 38 77 130)(13 17 41 39 78 131)(14 18 42 40 79 132)(15 19 43 33 80 133)(16 20 44 34 73 134)(57 122 87 117 111 91)(58 123 88 118 112 92)(59 124 81 119 105 93)(60 125 82 120 106 94)(61 126 83 113 107 95)(62 127 84 114 108 96)(63 128 85 115 109 89)(64 121 86 116 110 90)
(1 63 35 5 59 39)(2 64 36 6 60 40)(3 57 37 7 61 33)(4 58 38 8 62 34)(9 104 119 13 100 115)(10 97 120 14 101 116)(11 98 113 15 102 117)(12 99 114 16 103 118)(17 69 109 21 65 105)(18 70 110 22 66 106)(19 71 111 23 67 107)(20 72 112 24 68 108)(25 95 43 29 91 47)(26 96 44 30 92 48)(27 89 45 31 93 41)(28 90 46 32 94 42)(49 127 73 53 123 77)(50 128 74 54 124 78)(51 121 75 55 125 79)(52 122 76 56 126 80)(81 131 141 85 135 137)(82 132 142 86 136 138)(83 133 143 87 129 139)(84 134 144 88 130 140)
(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)
(1 103 104 8)(2 7 97 102)(3 101 98 6)(4 5 99 100)(9 62 39 118)(10 117 40 61)(11 60 33 116)(12 115 34 59)(13 58 35 114)(14 113 36 57)(15 64 37 120)(16 119 38 63)(17 92 74 84)(18 83 75 91)(19 90 76 82)(20 81 77 89)(21 96 78 88)(22 87 79 95)(23 94 80 86)(24 85 73 93)(25 66 143 51)(26 50 144 65)(27 72 137 49)(28 56 138 71)(29 70 139 55)(30 54 140 69)(31 68 141 53)(32 52 142 67)(41 112 135 127)(42 126 136 111)(43 110 129 125)(44 124 130 109)(45 108 131 123)(46 122 132 107)(47 106 133 121)(48 128 134 105)```

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 36 36 2 2 2 2 2 2 36 36 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D4 D6 D8 SD16 C4×S3 C3⋊D4 D12 C24⋊C2 D24 kernel C62.84D4 C12⋊Dic3 C6×C24 C2×C12⋊S3 C12⋊S3 C2×C24 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C6 C6 # reps 1 1 1 1 4 4 1 1 4 2 2 8 8 8 16 16

Matrix representation of C62.84D4 in GL4(𝔽73) generated by

 0 1 0 0 72 1 0 0 0 0 72 1 0 0 72 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 1
,
 43 60 0 0 13 30 0 0 0 0 25 11 0 0 62 36
,
 13 30 0 0 43 60 0 0 0 0 62 36 0 0 25 11
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[43,13,0,0,60,30,0,0,0,0,25,62,0,0,11,36],[13,43,0,0,30,60,0,0,0,0,62,25,0,0,36,11] >;`

C62.84D4 in GAP, Magma, Sage, TeX

`C_6^2._{84}D_4`
`% in TeX`

`G:=Group("C6^2.84D4");`
`// GroupNames label`

`G:=SmallGroup(288,296);`
`// by ID`

`G=gap.SmallGroup(288,296);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,422,100,2693,9414]);`
`// Polycyclic`

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

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