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## G = C32×SD32order 288 = 25·32

### Direct product of C32 and SD32

direct product, metacyclic, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C32×SD32
 Chief series C1 — C2 — C4 — C8 — C24 — C3×C24 — C32×Q16 — C32×SD32
 Lower central C1 — C2 — C4 — C8 — C32×SD32
 Upper central C1 — C3×C6 — C3×C12 — C3×C24 — C32×SD32

Generators and relations for C32×SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 168 in 78 conjugacy classes, 48 normal (16 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, C6 [×4], C6 [×4], C8, D4, Q8, C32, C12 [×4], C12 [×4], C2×C6 [×4], C16, D8, Q16, C3×C6, C3×C6, C24 [×4], C3×D4 [×4], C3×Q8 [×4], SD32, C3×C12, C3×C12, C62, C48 [×4], C3×D8 [×4], C3×Q16 [×4], C3×C24, D4×C32, Q8×C32, C3×SD32 [×4], C3×C48, C32×D8, C32×Q16, C32×SD32
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], D4, C32, C2×C6 [×4], D8, C3×C6 [×3], C3×D4 [×4], SD32, C62, C3×D8 [×4], D4×C32, C3×SD32 [×4], C32×D8, C32×SD32

Smallest permutation representation of C32×SD32
On 144 points
Generators in S144
(1 112 116)(2 97 117)(3 98 118)(4 99 119)(5 100 120)(6 101 121)(7 102 122)(8 103 123)(9 104 124)(10 105 125)(11 106 126)(12 107 127)(13 108 128)(14 109 113)(15 110 114)(16 111 115)(17 72 38)(18 73 39)(19 74 40)(20 75 41)(21 76 42)(22 77 43)(23 78 44)(24 79 45)(25 80 46)(26 65 47)(27 66 48)(28 67 33)(29 68 34)(30 69 35)(31 70 36)(32 71 37)(49 89 143)(50 90 144)(51 91 129)(52 92 130)(53 93 131)(54 94 132)(55 95 133)(56 96 134)(57 81 135)(58 82 136)(59 83 137)(60 84 138)(61 85 139)(62 86 140)(63 87 141)(64 88 142)
(1 51 40)(2 52 41)(3 53 42)(4 54 43)(5 55 44)(6 56 45)(7 57 46)(8 58 47)(9 59 48)(10 60 33)(11 61 34)(12 62 35)(13 63 36)(14 64 37)(15 49 38)(16 50 39)(17 110 89)(18 111 90)(19 112 91)(20 97 92)(21 98 93)(22 99 94)(23 100 95)(24 101 96)(25 102 81)(26 103 82)(27 104 83)(28 105 84)(29 106 85)(30 107 86)(31 108 87)(32 109 88)(65 123 136)(66 124 137)(67 125 138)(68 126 139)(69 127 140)(70 128 141)(71 113 142)(72 114 143)(73 115 144)(74 116 129)(75 117 130)(76 118 131)(77 119 132)(78 120 133)(79 121 134)(80 122 135)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 21)(18 28)(20 26)(22 24)(23 31)(25 29)(30 32)(33 39)(34 46)(35 37)(36 44)(38 42)(41 47)(43 45)(49 53)(50 60)(52 58)(54 56)(55 63)(57 61)(62 64)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)(81 85)(82 92)(84 90)(86 88)(87 95)(89 93)(94 96)(97 103)(98 110)(99 101)(100 108)(102 106)(105 111)(107 109)(113 127)(114 118)(115 125)(117 123)(119 121)(120 128)(122 126)(130 136)(131 143)(132 134)(133 141)(135 139)(138 144)(140 142)

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

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

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

99 conjugacy classes

 class 1 2A 2B 3A ··· 3H 4A 4B 6A ··· 6H 6I ··· 6P 8A 8B 12A ··· 12H 12I ··· 12P 16A 16B 16C 16D 24A ··· 24P 48A ··· 48AF order 1 2 2 3 ··· 3 4 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 16 16 16 16 24 ··· 24 48 ··· 48 size 1 1 8 1 ··· 1 2 8 1 ··· 1 8 ··· 8 2 2 2 ··· 2 8 ··· 8 2 2 2 2 2 ··· 2 2 ··· 2

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D8 C3×D4 SD32 C3×D8 C3×SD32 kernel C32×SD32 C3×C48 C32×D8 C32×Q16 C3×SD32 C48 C3×D8 C3×Q16 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 8 8 8 8 1 2 8 4 16 32

Matrix representation of C32×SD32 in GL3(𝔽97) generated by

 61 0 0 0 1 0 0 0 1
,
 35 0 0 0 61 0 0 0 61
,
 1 0 0 0 44 87 0 10 44
,
 96 0 0 0 1 0 0 0 96
G:=sub<GL(3,GF(97))| [61,0,0,0,1,0,0,0,1],[35,0,0,0,61,0,0,0,61],[1,0,0,0,44,10,0,87,44],[96,0,0,0,1,0,0,0,96] >;

C32×SD32 in GAP, Magma, Sage, TeX

C_3^2\times {\rm SD}_{32}
% in TeX

G:=Group("C3^2xSD32");
// GroupNames label

G:=SmallGroup(288,330);
// by ID

G=gap.SmallGroup(288,330);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,1008,533,3784,1901,242,9077,4548,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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