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## G = C62.227C23order 288 = 25·32

### 72nd non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.227C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C62.227C23
 Lower central C32 — C62 — C62.227C23
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C62.227C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=b3d >

Subgroups: 908 in 234 conjugacy classes, 67 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C3 [×4], C4 [×5], C22, C22 [×7], S3 [×8], C6 [×12], C6 [×4], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×12], C12 [×8], D6 [×16], C2×C6 [×4], C2×C6 [×12], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3×C6, C4×S3 [×8], C2×Dic3 [×12], C3⋊D4 [×8], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C22.D4, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×4], D6⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C2×C3⋊D4 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3 [×3], C327D4 [×2], C6×C12 [×2], C22×C3⋊S3, C2×C62, C23.9D6 [×4], C6.Dic6, C12⋊Dic3, C6.11D12, C625C4, C32×C22⋊C4, C2×C4×C3⋊S3, C2×C327D4, C62.227C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, C22×S3 [×4], C22.D4, C2×C3⋊S3 [×3], C4○D12 [×4], S3×D4 [×4], D42S3 [×4], C22×C3⋊S3, C23.9D6 [×4], C12.59D6, D4×C3⋊S3, C12.D6, C62.227C23

Smallest permutation representation of C62.227C23
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 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 17)(14 16)(19 141)(20 140)(21 139)(22 144)(23 143)(24 142)(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 46)(44 45)(47 48)(49 54)(50 53)(51 52)(67 101)(68 100)(69 99)(70 98)(71 97)(72 102)(73 93)(74 92)(75 91)(76 96)(77 95)(78 94)(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)(115 118)(116 117)(119 120)(121 126)(122 125)(123 124)
(1 79 4 82)(2 80 5 83)(3 81 6 84)(7 74 10 77)(8 75 11 78)(9 76 12 73)(13 90 16 87)(14 85 17 88)(15 86 18 89)(19 99 22 102)(20 100 23 97)(21 101 24 98)(25 95 28 92)(26 96 29 93)(27 91 30 94)(31 106 34 103)(32 107 35 104)(33 108 36 105)(37 112 40 109)(38 113 41 110)(39 114 42 111)(43 118 46 115)(44 119 47 116)(45 120 48 117)(49 124 52 121)(50 125 53 122)(51 126 54 123)(55 130 58 127)(56 131 59 128)(57 132 60 129)(61 136 64 133)(62 137 65 134)(63 138 66 135)(67 142 70 139)(68 143 71 140)(69 144 72 141)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 51)(14 52)(15 53)(16 54)(17 49)(18 50)(19 132)(20 127)(21 128)(22 129)(23 130)(24 131)(25 134)(26 135)(27 136)(28 137)(29 138)(30 133)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(79 125)(80 126)(81 121)(82 122)(83 123)(84 124)(85 117)(86 118)(87 119)(88 120)(89 115)(90 116)(109 144)(110 139)(111 140)(112 141)(113 142)(114 143)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C4○D12 S3×D4 D4⋊2S3 kernel C62.227C23 C6.Dic6 C12⋊Dic3 C6.11D12 C62⋊5C4 C32×C22⋊C4 C2×C4×C3⋊S3 C2×C32⋊7D4 C3×C22⋊C4 C2×C3⋊S3 C2×C12 C22×C6 C3×C6 C6 C6 C6 # reps 1 1 1 1 1 1 1 1 4 2 8 4 4 16 4 4

Matrix representation of C62.227C23 in GL6(𝔽13)

 1 1 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 4 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 7 8
,
 2 4 0 0 0 0 9 11 0 0 0 0 0 0 11 9 0 0 0 0 4 2 0 0 0 0 0 0 7 3 0 0 0 0 10 6

`G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,4,0,0,0,0,0,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,7,0,0,0,0,0,8],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,7,10,0,0,0,0,3,6] >;`

C62.227C23 in GAP, Magma, Sage, TeX

`C_6^2._{227}C_2^3`
`% in TeX`

`G:=Group("C6^2.227C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,740);`
`// by ID`

`G=gap.SmallGroup(288,740);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,2693,9414]);`
`// Polycyclic`

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

׿
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