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## G = C62.228C23order 288 = 25·32

### 73rd non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.228C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C62.228C23
 Lower central C32 — C62 — C62.228C23
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C62.228C23
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=a3b3c, ede=b3d >

Subgroups: 1260 in 282 conjugacy classes, 69 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×5], C22, C22 [×10], S3 [×12], C6 [×12], C6 [×4], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×12], C12 [×8], D6 [×28], C2×C6 [×4], C2×C6 [×12], C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×3], C3×C6 [×3], C3×C6, C4×S3 [×8], D12 [×8], C2×Dic3 [×8], C3⋊D4 [×16], C2×C12 [×8], C22×S3 [×8], C22×C6 [×4], C4⋊D4, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×5], C62, C62 [×3], Dic3⋊C4 [×4], D6⋊C4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C2×D12 [×4], C2×C3⋊D4 [×8], C4×C3⋊S3 [×2], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12 [×2], C22×C3⋊S3 [×2], C2×C62, Dic3⋊D4 [×4], C6.Dic6, C6.11D12, C32×C22⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C327D4 [×2], C62.228C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], C4○D12 [×4], S3×D4 [×8], C22×C3⋊S3, Dic3⋊D4 [×4], C12.59D6, D4×C3⋊S3 [×2], C62.228C23

Smallest permutation representation of C62.228C23
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 18 58 31 39 61)(2 13 59 32 40 62)(3 14 60 33 41 63)(4 15 55 34 42 64)(5 16 56 35 37 65)(6 17 57 36 38 66)(7 126 27 24 140 118)(8 121 28 19 141 119)(9 122 29 20 142 120)(10 123 30 21 143 115)(11 124 25 22 144 116)(12 125 26 23 139 117)(43 92 51 73 101 71)(44 93 52 74 102 72)(45 94 53 75 97 67)(46 95 54 76 98 68)(47 96 49 77 99 69)(48 91 50 78 100 70)(79 128 87 109 137 107)(80 129 88 110 138 108)(81 130 89 111 133 103)(82 131 90 112 134 104)(83 132 85 113 135 105)(84 127 86 114 136 106)
(1 139)(2 144)(3 143)(4 142)(5 141)(6 140)(7 38)(8 37)(9 42)(10 41)(11 40)(12 39)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(25 59)(26 58)(27 57)(28 56)(29 55)(30 60)(31 125)(32 124)(33 123)(34 122)(35 121)(36 126)(43 138)(44 137)(45 136)(46 135)(47 134)(48 133)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(61 117)(62 116)(63 115)(64 120)(65 119)(66 118)(67 106)(68 105)(69 104)(70 103)(71 108)(72 107)(73 129)(74 128)(75 127)(76 132)(77 131)(78 130)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(91 111)(92 110)(93 109)(94 114)(95 113)(96 112)
(1 106 4 103)(2 107 5 104)(3 108 6 105)(7 95 10 92)(8 96 11 93)(9 91 12 94)(13 79 16 82)(14 80 17 83)(15 81 18 84)(19 99 22 102)(20 100 23 97)(21 101 24 98)(25 74 28 77)(26 75 29 78)(27 76 30 73)(31 86 34 89)(32 87 35 90)(33 88 36 85)(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 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 134)(8 135)(9 136)(10 137)(11 138)(12 133)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 132)(20 127)(21 128)(22 129)(23 130)(24 131)(25 80)(26 81)(27 82)(28 83)(29 84)(30 79)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(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)(85 141)(86 142)(87 143)(88 144)(89 139)(90 140)(103 125)(104 126)(105 121)(106 122)(107 123)(108 124)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C4○D12 S3×D4 kernel C62.228C23 C6.Dic6 C6.11D12 C32×C22⋊C4 C2×C4×C3⋊S3 C2×C12⋊S3 C2×C32⋊7D4 C3×C22⋊C4 C3⋊Dic3 C2×C3⋊S3 C2×C12 C22×C6 C3×C6 C6 C6 # reps 1 1 1 1 1 1 2 4 2 2 8 4 2 16 8

Matrix representation of C62.228C23 in GL6(𝔽13)

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

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

C62.228C23 in GAP, Magma, Sage, TeX

`C_6^2._{228}C_2^3`
`% in TeX`

`G:=Group("C6^2.228C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,741);`
`// by ID`

`G=gap.SmallGroup(288,741);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,2693,9414]);`
`// 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*b^3*c,e*d*e=b^3*d>;`
`// generators/relations`

׿
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