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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 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6, C3×C6, C24 [×8], C2×C12 [×4], M5(2), C3×C12 [×2], C62, C48 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C3×M5(2) [×4], C3×C48 [×2], C6×C24, C32×M5(2)
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, C3×C6 [×3], C24 [×8], C2×C12 [×4], M5(2), C3×C12 [×2], C62, C2×C24 [×4], C3×C24 [×2], C6×C12, C3×M5(2) [×4], C6×C24, C32×M5(2)

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

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

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

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

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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