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G = C32×M5(2)  order 288 = 25·32

Direct product of C32 and M5(2)

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C32×M5(2)
 Chief series C1 — C2 — C4 — C8 — C24 — C3×C24 — C3×C48 — C32×M5(2)
 Lower central C1 — C2 — C32×M5(2)
 Upper central C1 — C3×C24 — C32×M5(2)

Generators and relations for C32×M5(2)
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >

Subgroups: 84 in 78 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, C16, C2×C8, C3×C6, C3×C6, C24, C2×C12, M5(2), C3×C12, C62, C48, C2×C24, C3×C24, C6×C12, C3×M5(2), C3×C48, C6×C24, C32×M5(2)
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, M5(2), C3×C12, C62, C2×C24, C3×C24, C6×C12, C3×M5(2), C6×C24, C32×M5(2)

Smallest permutation representation of C32×M5(2)
On 144 points
Generators in S144
(1 34 90)(2 35 91)(3 36 92)(4 37 93)(5 38 94)(6 39 95)(7 40 96)(8 41 81)(9 42 82)(10 43 83)(11 44 84)(12 45 85)(13 46 86)(14 47 87)(15 48 88)(16 33 89)(17 101 68)(18 102 69)(19 103 70)(20 104 71)(21 105 72)(22 106 73)(23 107 74)(24 108 75)(25 109 76)(26 110 77)(27 111 78)(28 112 79)(29 97 80)(30 98 65)(31 99 66)(32 100 67)(49 113 143)(50 114 144)(51 115 129)(52 116 130)(53 117 131)(54 118 132)(55 119 133)(56 120 134)(57 121 135)(58 122 136)(59 123 137)(60 124 138)(61 125 139)(62 126 140)(63 127 141)(64 128 142)
(1 68 118)(2 69 119)(3 70 120)(4 71 121)(5 72 122)(6 73 123)(7 74 124)(8 75 125)(9 76 126)(10 77 127)(11 78 128)(12 79 113)(13 80 114)(14 65 115)(15 66 116)(16 67 117)(17 132 34)(18 133 35)(19 134 36)(20 135 37)(21 136 38)(22 137 39)(23 138 40)(24 139 41)(25 140 42)(26 141 43)(27 142 44)(28 143 45)(29 144 46)(30 129 47)(31 130 48)(32 131 33)(49 85 112)(50 86 97)(51 87 98)(52 88 99)(53 89 100)(54 90 101)(55 91 102)(56 92 103)(57 93 104)(58 94 105)(59 95 106)(60 96 107)(61 81 108)(62 82 109)(63 83 110)(64 84 111)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)(113 121)(115 123)(117 125)(119 127)(129 137)(131 139)(133 141)(135 143)

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

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

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

180 conjugacy classes

 class 1 2A 2B 3A ··· 3H 4A 4B 4C 6A ··· 6H 6I ··· 6P 8A 8B 8C 8D 8E 8F 12A ··· 12P 12Q ··· 12X 16A ··· 16H 24A ··· 24AF 24AG ··· 24AV 48A ··· 48BL order 1 2 2 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 8 8 12 ··· 12 12 ··· 12 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 2 1 ··· 1 1 1 2 1 ··· 1 2 ··· 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C24 C24 M5(2) C3×M5(2) kernel C32×M5(2) C3×C48 C6×C24 C3×M5(2) C3×C24 C6×C12 C48 C2×C24 C3×C12 C62 C24 C2×C12 C12 C2×C6 C32 C3 # reps 1 2 1 8 2 2 16 8 4 4 16 16 32 32 4 32

Matrix representation of C32×M5(2) in GL3(𝔽97) generated by

 61 0 0 0 61 0 0 0 61
,
 61 0 0 0 35 0 0 0 35
,
 1 0 0 0 22 95 0 80 75
,
 96 0 0 0 1 0 0 22 96
G:=sub<GL(3,GF(97))| [61,0,0,0,61,0,0,0,61],[61,0,0,0,35,0,0,0,35],[1,0,0,0,22,80,0,95,75],[96,0,0,0,1,22,0,0,96] >;

C32×M5(2) in GAP, Magma, Sage, TeX

C_3^2\times M_5(2)
% in TeX

G:=Group("C3^2xM5(2)");
// GroupNames label

G:=SmallGroup(288,328);
// by ID

G=gap.SmallGroup(288,328);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,252,2045,102,124]);
// 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^9>;
// generators/relations

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