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## G = C62.229C23order 288 = 25·32

### 74th non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.229C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C6.11D12 — C62.229C23
 Lower central C32 — C62 — C62.229C23
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C62.229C23
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, dcd-1=b3c, ece=a3b3c, ede=b3d >

Subgroups: 908 in 228 conjugacy classes, 67 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×12], C6 [×4], C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, C23, C32, Dic3 [×16], C12 [×8], D6 [×12], C2×C6 [×4], C2×C6 [×12], C42, C22⋊C4, C22⋊C4 [×3], C2×D4, C2×Q8, C3⋊S3, C3×C6 [×3], C3×C6, Dic6 [×8], C2×Dic3 [×12], C3⋊D4 [×8], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C4.4D4, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×3], C62, C62 [×3], C4×Dic3 [×4], D6⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×4], C2×Dic6 [×4], C2×C3⋊D4 [×4], C324Q8 [×2], C2×C3⋊Dic3 [×3], C327D4 [×2], C6×C12 [×2], C22×C3⋊S3, C2×C62, C23.11D6 [×4], C4×C3⋊Dic3, C6.11D12 [×2], C625C4, C32×C22⋊C4, C2×C324Q8, C2×C327D4, C62.229C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, C22×S3 [×4], C4.4D4, C2×C3⋊S3 [×3], C4○D12 [×4], S3×D4 [×4], D42S3 [×4], C22×C3⋊S3, C23.11D6 [×4], C12.59D6, D4×C3⋊S3, C12.D6, C62.229C23

Smallest permutation representation of C62.229C23
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 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 22 123 116 29 140)(8 23 124 117 30 141)(9 24 125 118 25 142)(10 19 126 119 26 143)(11 20 121 120 27 144)(12 21 122 115 28 139)(43 92 67 73 101 50)(44 93 68 74 102 51)(45 94 69 75 97 52)(46 95 70 76 98 53)(47 96 71 77 99 54)(48 91 72 78 100 49)(79 128 103 109 137 86)(80 129 104 110 138 87)(81 130 105 111 133 88)(82 131 106 112 134 89)(83 132 107 113 135 90)(84 127 108 114 136 85)
(2 6)(3 5)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(19 118)(20 117)(21 116)(22 115)(23 120)(24 119)(31 35)(32 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 93)(44 92)(45 91)(46 96)(47 95)(48 94)(49 69)(50 68)(51 67)(52 72)(53 71)(54 70)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(121 124)(122 123)(125 126)(139 140)(141 144)(142 143)
(1 85 4 88)(2 86 5 89)(3 87 6 90)(7 74 10 77)(8 75 11 78)(9 76 12 73)(13 82 16 79)(14 83 17 80)(15 84 18 81)(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 52)(2 53)(3 54)(4 49)(5 50)(6 51)(7 80)(8 81)(9 82)(10 83)(11 84)(12 79)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(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)(85 144)(86 139)(87 140)(88 141)(89 142)(90 143)(103 122)(104 123)(105 124)(106 125)(107 126)(108 121)(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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,22,123,116,29,140)(8,23,124,117,30,141)(9,24,125,118,25,142)(10,19,126,119,26,143)(11,20,121,120,27,144)(12,21,122,115,28,139)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(19,118)(20,117)(21,116)(22,115)(23,120)(24,119)(31,35)(32,34)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,93)(44,92)(45,91)(46,96)(47,95)(48,94)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(121,124)(122,123)(125,126)(139,140)(141,144)(142,143), (1,85,4,88)(2,86,5,89)(3,87,6,90)(7,74,10,77)(8,75,11,78)(9,76,12,73)(13,82,16,79)(14,83,17,80)(15,84,18,81)(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), (1,52)(2,53)(3,54)(4,49)(5,50)(6,51)(7,80)(8,81)(9,82)(10,83)(11,84)(12,79)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(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)(85,144)(86,139)(87,140)(88,141)(89,142)(90,143)(103,122)(104,123)(105,124)(106,125)(107,126)(108,121)(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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,22,123,116,29,140)(8,23,124,117,30,141)(9,24,125,118,25,142)(10,19,126,119,26,143)(11,20,121,120,27,144)(12,21,122,115,28,139)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(19,118)(20,117)(21,116)(22,115)(23,120)(24,119)(31,35)(32,34)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,93)(44,92)(45,91)(46,96)(47,95)(48,94)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(121,124)(122,123)(125,126)(139,140)(141,144)(142,143), (1,85,4,88)(2,86,5,89)(3,87,6,90)(7,74,10,77)(8,75,11,78)(9,76,12,73)(13,82,16,79)(14,83,17,80)(15,84,18,81)(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), (1,52)(2,53)(3,54)(4,49)(5,50)(6,51)(7,80)(8,81)(9,82)(10,83)(11,84)(12,79)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(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)(85,144)(86,139)(87,140)(88,141)(89,142)(90,143)(103,122)(104,123)(105,124)(106,125)(107,126)(108,121)(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)], [(1,15,58,33,39,61),(2,16,59,34,40,62),(3,17,60,35,41,63),(4,18,55,36,42,64),(5,13,56,31,37,65),(6,14,57,32,38,66),(7,22,123,116,29,140),(8,23,124,117,30,141),(9,24,125,118,25,142),(10,19,126,119,26,143),(11,20,121,120,27,144),(12,21,122,115,28,139),(43,92,67,73,101,50),(44,93,68,74,102,51),(45,94,69,75,97,52),(46,95,70,76,98,53),(47,96,71,77,99,54),(48,91,72,78,100,49),(79,128,103,109,137,86),(80,129,104,110,138,87),(81,130,105,111,133,88),(82,131,106,112,134,89),(83,132,107,113,135,90),(84,127,108,114,136,85)], [(2,6),(3,5),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,63),(14,62),(15,61),(16,66),(17,65),(18,64),(19,118),(20,117),(21,116),(22,115),(23,120),(24,119),(31,35),(32,34),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,93),(44,92),(45,91),(46,96),(47,95),(48,94),(49,69),(50,68),(51,67),(52,72),(53,71),(54,70),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,132),(80,131),(81,130),(82,129),(83,128),(84,127),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(109,135),(110,134),(111,133),(112,138),(113,137),(114,136),(121,124),(122,123),(125,126),(139,140),(141,144),(142,143)], [(1,85,4,88),(2,86,5,89),(3,87,6,90),(7,74,10,77),(8,75,11,78),(9,76,12,73),(13,82,16,79),(14,83,17,80),(15,84,18,81),(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)], [(1,52),(2,53),(3,54),(4,49),(5,50),(6,51),(7,80),(8,81),(9,82),(10,83),(11,84),(12,79),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(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),(85,144),(86,139),(87,140),(88,141),(89,142),(90,143),(103,122),(104,123),(105,124),(106,125),(107,126),(108,121),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120)])`

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 6M ··· 6T 12A ··· 12P order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 4 36 2 2 2 2 2 2 4 18 18 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 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C4○D12 S3×D4 D4⋊2S3 kernel C62.229C23 C4×C3⋊Dic3 C6.11D12 C62⋊5C4 C32×C22⋊C4 C2×C32⋊4Q8 C2×C32⋊7D4 C3×C22⋊C4 C3⋊Dic3 C2×C12 C22×C6 C3×C6 C6 C6 C6 # reps 1 1 2 1 1 1 1 4 2 8 4 4 16 4 4

Matrix representation of C62.229C23 in GL6(𝔽13)

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

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

C62.229C23 in GAP, Magma, Sage, TeX

`C_6^2._{229}C_2^3`
`% in TeX`

`G:=Group("C6^2.229C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,742);`
`// by ID`

`G=gap.SmallGroup(288,742);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,2693,9414]);`
`// 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,d*c*d^-1=b^3*c,e*c*e=a^3*b^3*c,e*d*e=b^3*d>;`
`// generators/relations`

׿
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