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## G = C62.223C23order 288 = 25·32

### 68th non-split extension by C62 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.223C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.223C23
 Lower central C32 — C62 — C62.223C23
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C62.223C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=b3d >

Subgroups: 556 in 180 conjugacy classes, 65 normal (29 characteristic)
C1, C2 [×3], C2, C3 [×4], C4 [×6], C22, C22 [×3], C6 [×12], C6 [×4], C2×C4 [×2], C2×C4 [×4], C23, C32, Dic3 [×16], C12 [×8], C2×C6 [×4], C2×C6 [×12], C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C3×C6 [×3], C3×C6, C2×Dic3 [×16], C2×C12 [×8], C22×C6 [×4], C422C2, C3⋊Dic3 [×4], C3×C12 [×2], C62, C62 [×3], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×8], C3×C22⋊C4 [×4], C2×C3⋊Dic3 [×4], C6×C12 [×2], C2×C62, C23.8D6 [×4], C4×C3⋊Dic3, C6.Dic6 [×2], C12⋊Dic3, C625C4 [×2], C32×C22⋊C4, C62.223C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4 [×3], C3⋊S3, C22×S3 [×4], C422C2, C2×C3⋊S3 [×3], C4○D12 [×4], D42S3 [×8], C22×C3⋊S3, C23.8D6 [×4], C12.59D6, C12.D6 [×2], C62.223C23

Smallest permutation representation of C62.223C23
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 31 38 57 16 52)(2 32 39 58 17 53)(3 33 40 59 18 54)(4 34 41 60 13 49)(5 35 42 55 14 50)(6 36 37 56 15 51)(7 142 27 22 123 137)(8 143 28 23 124 138)(9 144 29 24 125 133)(10 139 30 19 126 134)(11 140 25 20 121 135)(12 141 26 21 122 136)(43 93 73 63 83 71)(44 94 74 64 84 72)(45 95 75 65 79 67)(46 96 76 66 80 68)(47 91 77 61 81 69)(48 92 78 62 82 70)(85 99 112 104 132 118)(86 100 113 105 127 119)(87 101 114 106 128 120)(88 102 109 107 129 115)(89 97 110 108 130 116)(90 98 111 103 131 117)
(1 100 57 127)(2 99 58 132)(3 98 59 131)(4 97 60 130)(5 102 55 129)(6 101 56 128)(7 94 22 84)(8 93 23 83)(9 92 24 82)(10 91 19 81)(11 96 20 80)(12 95 21 79)(13 108 34 89)(14 107 35 88)(15 106 36 87)(16 105 31 86)(17 104 32 85)(18 103 33 90)(25 68 135 76)(26 67 136 75)(27 72 137 74)(28 71 138 73)(29 70 133 78)(30 69 134 77)(37 120 51 114)(38 119 52 113)(39 118 53 112)(40 117 54 111)(41 116 49 110)(42 115 50 109)(43 124 63 143)(44 123 64 142)(45 122 65 141)(46 121 66 140)(47 126 61 139)(48 125 62 144)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 114 10 111)(8 109 11 112)(9 110 12 113)(13 65 16 62)(14 66 17 63)(15 61 18 64)(19 117 22 120)(20 118 23 115)(21 119 24 116)(25 132 28 129)(26 127 29 130)(27 128 30 131)(31 48 34 45)(32 43 35 46)(33 44 36 47)(37 91 40 94)(38 92 41 95)(39 93 42 96)(49 79 52 82)(50 80 53 83)(51 81 54 84)(55 76 58 73)(56 77 59 74)(57 78 60 75)(85 124 88 121)(86 125 89 122)(87 126 90 123)(97 136 100 133)(98 137 101 134)(99 138 102 135)(103 142 106 139)(104 143 107 140)(105 144 108 141)
(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(43 63)(44 64)(45 65)(46 66)(47 61)(48 62)(67 75)(68 76)(69 77)(70 78)(71 73)(72 74)(79 95)(80 96)(81 91)(82 92)(83 93)(84 94)(85 107)(86 108)(87 103)(88 104)(89 105)(90 106)(97 127)(98 128)(99 129)(100 130)(101 131)(102 132)(109 118)(110 119)(111 120)(112 115)(113 116)(114 117)(121 124)(122 125)(123 126)(133 136)(134 137)(135 138)(139 142)(140 143)(141 144)

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 D4⋊2S3 kernel C62.223C23 C4×C3⋊Dic3 C6.Dic6 C12⋊Dic3 C62⋊5C4 C32×C22⋊C4 C3×C22⋊C4 C2×C12 C22×C6 C3×C6 C6 C6 # reps 1 1 2 1 2 1 4 8 4 6 16 8

Matrix representation of C62.223C23 in GL6(𝔽13)

 10 0 0 0 0 0 0 4 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 10 0 0 0 0 0 4 4
,
 3 0 0 0 0 0 0 9 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 8 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 12 0 0 0 0 2 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 10 12

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

C62.223C23 in GAP, Magma, Sage, TeX

C_6^2._{223}C_2^3
% in TeX

G:=Group("C6^2.223C2^3");
// GroupNames label

G:=SmallGroup(288,736);
// by ID

G=gap.SmallGroup(288,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,219,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=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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