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## G = C62.225C23order 288 = 25·32

### 70th non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.225C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C32⋊7D4 — C62.225C23
 Lower central C32 — C3×C6 — C62.225C23
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C62.225C23
G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd=b3c, ce=ec, ede-1=b3d >

Subgroups: 996 in 282 conjugacy classes, 87 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×7], C22, C22 [×2], C22 [×6], S3 [×8], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×20], C12 [×8], D6 [×16], C2×C6 [×12], C2×C6 [×8], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×20], C3⋊D4 [×16], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C4×D4, C3⋊Dic3 [×4], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3 [×4], C2×C3⋊D4 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3 [×3], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12 [×2], C22×C3⋊S3, C2×C62, Dic34D4 [×4], C4×C3⋊Dic3, C6.Dic6, C6.11D12, C32×C22⋊C4, C2×C4×C3⋊S3, C22×C3⋊Dic3, C2×C327D4, C62.225C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], C23, D6 [×12], C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C4×D4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], S3×D4 [×4], D42S3 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, Dic34D4 [×4], C2×C4×C3⋊S3, D4×C3⋊S3, C12.D6, C62.225C23

Smallest permutation representation of C62.225C23
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 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(25 116)(26 115)(27 120)(28 119)(29 118)(30 117)(31 35)(32 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 99)(44 98)(45 97)(46 102)(47 101)(48 100)(50 54)(51 53)(67 71)(68 70)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(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)(121 144)(122 143)(123 142)(124 141)(125 140)(126 139)
(1 88)(2 89)(3 90)(4 85)(5 86)(6 87)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 99)(20 100)(21 101)(22 102)(23 97)(24 98)(25 95)(26 96)(27 91)(28 92)(29 93)(30 94)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 49 4 52)(2 50 5 53)(3 51 6 54)(7 80 10 83)(8 81 11 84)(9 82 12 79)(13 46 16 43)(14 47 17 44)(15 48 18 45)(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)(85 141 88 144)(86 142 89 139)(87 143 90 140)(103 125 106 122)(104 126 107 123)(105 121 108 124)(109 118 112 115)(110 119 113 116)(111 120 114 117)```

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 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 C4 S3 D4 D6 D6 C4○D4 C4×S3 S3×D4 D4⋊2S3 kernel C62.225C23 C4×C3⋊Dic3 C6.Dic6 C6.11D12 C32×C22⋊C4 C2×C4×C3⋊S3 C22×C3⋊Dic3 C2×C32⋊7D4 C32⋊7D4 C3×C22⋊C4 C3⋊Dic3 C2×C12 C22×C6 C3×C6 C2×C6 C6 C6 # reps 1 1 1 1 1 1 1 1 8 4 2 8 4 2 16 4 4

Matrix representation of C62.225C23 in GL6(𝔽13)

 12 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 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 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 0 0 0 0 0 12 12 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 1 11 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 1 12

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

C62.225C23 in GAP, Magma, Sage, TeX

`C_6^2._{225}C_2^3`
`% in TeX`

`G:=Group("C6^2.225C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,738);`
`// by ID`

`G=gap.SmallGroup(288,738);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,2693,9414]);`
`// Polycyclic`

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

׿
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