direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C32×M5(2), C48⋊7C6, C12.6C24, C62.5C8, C8.8C62, C24.11C12, C4.(C3×C24), C16⋊3(C3×C6), (C3×C48)⋊11C2, C8.2(C3×C12), (C2×C6).4C24, C2.3(C6×C24), C22.(C3×C24), (C2×C24).33C6, (C6×C24).28C2, (C6×C12).34C4, C24.42(C2×C6), (C3×C24).15C4, (C3×C12).13C8, C6.16(C2×C24), C4.10(C6×C12), C12.62(C2×C12), (C2×C12).24C12, (C3×C24).76C22, (C2×C8).8(C3×C6), (C2×C4).5(C3×C12), (C3×C6).49(C2×C8), (C3×C12).146(C2×C4), SmallGroup(288,328)
Series: Derived ►Chief ►Lower central ►Upper central
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)
►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