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## G = C62.74D4order 288 = 25·32

### 58th non-split extension by C62 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.74D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C12.59D6 — C62.74D4
 Lower central C32 — C3×C6 — C3×C12 — C62.74D4
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for C62.74D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a-1, dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c3 >

Subgroups: 644 in 186 conjugacy classes, 65 normal (29 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], S3 [×4], C6 [×4], C6 [×8], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C32, Dic3 [×4], C12 [×8], C12 [×4], D6 [×4], C2×C6 [×4], C2×C6 [×4], C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊S3, C3×C6, C3×C6 [×2], C3⋊C8 [×8], Dic6 [×4], C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C12 [×4], C2×C12 [×4], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×4], C4○D8, C3⋊Dic3, C3×C12 [×2], C3×C12, C2×C3⋊S3, C62, C62, C2×C3⋊C8 [×4], D4⋊S3 [×4], D4.S3 [×4], Q82S3 [×4], C3⋊Q16 [×4], C4○D12 [×4], C3×C4○D4 [×4], C324C8 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.13D6 [×4], C2×C324C8, C327D8, C329SD16, C3211SD16, C327Q16, C12.59D6, C32×C4○D4, C62.74D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C3⋊D4 [×8], C22×S3 [×4], C4○D8, C2×C3⋊S3 [×3], C2×C3⋊D4 [×4], C327D4 [×2], C22×C3⋊S3, Q8.13D6 [×4], C2×C327D4, C62.74D4

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E ··· 6P 8A 8B 8C 8D 12A ··· 12H 12I ··· 12T order 1 2 2 2 2 3 3 3 3 4 4 4 4 4 6 6 6 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 size 1 1 2 4 36 2 2 2 2 1 1 2 4 36 2 2 2 2 4 ··· 4 18 18 18 18 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C4○D8 Q8.13D6 kernel C62.74D4 C2×C32⋊4C8 C32⋊7D8 C32⋊9SD16 C32⋊11SD16 C32⋊7Q16 C12.59D6 C32×C4○D4 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C12 C2×C6 C32 C3 # reps 1 1 1 1 1 1 1 1 4 1 1 4 4 4 8 8 4 8

Matrix representation of C62.74D4 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 46 19 0 0 0 0 27 27
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 60 43 0 0 0 0 30 13 0 0 0 0 0 0 30 60 0 0 0 0 30 43 0 0 0 0 0 0 32 32 0 0 0 0 57 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72

`G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,19,27],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[60,30,0,0,0,0,43,13,0,0,0,0,0,0,30,30,0,0,0,0,60,43,0,0,0,0,0,0,32,57,0,0,0,0,32,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;`

C62.74D4 in GAP, Magma, Sage, TeX

`C_6^2._{74}D_4`
`% in TeX`

`G:=Group("C6^2.74D4");`
`// GroupNames label`

`G:=SmallGroup(288,807);`
`// by ID`

`G=gap.SmallGroup(288,807);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,675,185,80,2693,9414]);`
`// Polycyclic`

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

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