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## G = C62.236C23order 288 = 25·32

### 81st non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.236C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C62.236C23
 Lower central C32 — C3×C6 — C62.236C23
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C62.236C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 772 in 228 conjugacy classes, 85 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C32, Dic3 [×16], C12 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4, C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×16], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C42⋊C2, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3 [×8], C4⋊Dic3 [×4], D6⋊C4 [×8], C3×C4⋊C4 [×4], S3×C2×C4 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3, C4⋊C47S3 [×4], C4×C3⋊Dic3 [×2], C12⋊Dic3, C6.11D12 [×2], C32×C4⋊C4, C2×C4×C3⋊S3, C62.236C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C4○D4 [×2], C3⋊S3, C4×S3 [×8], C22×S3 [×4], C42⋊C2, C2×C3⋊S3 [×3], S3×C2×C4 [×4], D42S3 [×4], Q83S3 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, C4⋊C47S3 [×4], C2×C4×C3⋊S3, C12.D6, C12.26D6, C62.236C23

Smallest permutation representation of C62.236C23
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 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)
(2 6)(3 5)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 17)(14 16)(25 139)(26 144)(27 143)(28 142)(29 141)(30 140)(31 62)(32 61)(33 66)(34 65)(35 64)(36 63)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 53)(44 52)(45 51)(46 50)(47 49)(48 54)(67 95)(68 94)(69 93)(70 92)(71 91)(72 96)(73 99)(74 98)(75 97)(76 102)(77 101)(78 100)(80 84)(81 83)(85 87)(88 90)(103 134)(104 133)(105 138)(106 137)(107 136)(108 135)(109 132)(110 131)(111 130)(112 129)(113 128)(114 127)(115 125)(116 124)(117 123)(118 122)(119 121)(120 126)
(1 86 18 82)(2 87 13 83)(3 88 14 84)(4 89 15 79)(5 90 16 80)(6 85 17 81)(7 74 139 67)(8 75 140 68)(9 76 141 69)(10 77 142 70)(11 78 143 71)(12 73 144 72)(19 99 26 96)(20 100 27 91)(21 101 28 92)(22 102 29 93)(23 97 30 94)(24 98 25 95)(31 110 38 103)(32 111 39 104)(33 112 40 105)(34 113 41 106)(35 114 42 107)(36 109 37 108)(43 125 53 115)(44 126 54 116)(45 121 49 117)(46 122 50 118)(47 123 51 119)(48 124 52 120)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 133 61 130)(59 134 62 131)(60 135 63 132)
(1 43 4 46)(2 44 5 47)(3 45 6 48)(7 103 10 106)(8 104 11 107)(9 105 12 108)(13 54 16 51)(14 49 17 52)(15 50 18 53)(19 132 22 129)(20 127 23 130)(21 128 24 131)(25 134 28 137)(26 135 29 138)(27 136 30 133)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 76 40 73)(38 77 41 74)(39 78 42 75)(55 94 58 91)(56 95 59 92)(57 96 60 93)(61 100 64 97)(62 101 65 98)(63 102 66 99)(79 122 82 125)(80 123 83 126)(81 124 84 121)(85 120 88 117)(86 115 89 118)(87 116 90 119)(109 141 112 144)(110 142 113 139)(111 143 114 140)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A ··· 6L 12A ··· 12X order 1 2 2 2 2 2 3 3 3 3 4 ··· 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 2 2 2 2 2 ··· 2 9 9 9 9 18 18 18 18 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 D6 C4○D4 C4×S3 D4⋊2S3 Q8⋊3S3 kernel C62.236C23 C4×C3⋊Dic3 C12⋊Dic3 C6.11D12 C32×C4⋊C4 C2×C4×C3⋊S3 C4×C3⋊S3 C3×C4⋊C4 C2×C12 C3×C6 C12 C6 C6 # reps 1 2 1 2 1 1 8 4 12 4 16 4 4

Matrix representation of C62.236C23 in GL6(𝔽13)

 12 3 0 0 0 0 12 2 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 3 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 8 5
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 11 0 0 0 0 1 12

`G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,3,2,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,3,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,11,12] >;`

C62.236C23 in GAP, Magma, Sage, TeX

`C_6^2._{236}C_2^3`
`% in TeX`

`G:=Group("C6^2.236C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,749);`
`// by ID`

`G=gap.SmallGroup(288,749);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,219,58,2693,9414]);`
`// Polycyclic`

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

׿
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