metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊5M4(2), Dic5⋊4M4(2), C42.203D10, C4⋊C8⋊13D5, C4.55(Q8×D5), C4⋊3(C8⋊D5), C20⋊3C8⋊14C2, (C4×D5).72D4, C4.207(D4×D5), (C4×D5).18Q8, C20.366(C2×D4), (C2×C8).182D10, C20.113(C2×Q8), (D5×C42).4C2, D10.12(C4⋊C4), (C4×C20).63C22, C5⋊4(C4⋊M4(2)), C20.8Q8⋊23C2, (C4×Dic5).10C4, C2.18(D5×M4(2)), Dic5.13(C4⋊C4), (C2×C20).834C23, (C2×C40).213C22, C10.42(C2×M4(2)), (C4×Dic5).308C22, (C5×C4⋊C8)⋊18C2, C2.9(D5×C4⋊C4), (C2×C4×D5).11C4, C10.31(C2×C4⋊C4), (C2×C4).147(C4×D5), C2.13(C2×C8⋊D5), C22.112(C2×C4×D5), (C2×C20).244(C2×C4), (C2×C8⋊D5).12C2, (C2×C4×D5).350C22, (C2×C4).776(C22×D5), (C2×C10).190(C22×C4), (C2×C5⋊2C8).200C22, (C2×Dic5).145(C2×C4), (C22×D5).103(C2×C4), SmallGroup(320,464)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊5M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a9, cbc=b5 >
Subgroups: 398 in 126 conjugacy classes, 61 normal (37 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, C20, D10, D10, C2×C10, C4⋊C8, C4⋊C8, C2×C42, C2×M4(2), C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C4⋊M4(2), C8⋊D5, C2×C5⋊2C8, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C20⋊3C8, C20.8Q8, C5×C4⋊C8, D5×C42, C2×C8⋊D5, C20⋊5M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C2×M4(2), C4×D5, C22×D5, C4⋊M4(2), C8⋊D5, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, C2×C8⋊D5, D5×M4(2), C20⋊5M4(2)
►On 160 points
Generators in S160
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 24 136 152 69 56 100 105)(2 23 137 151 70 55 81 104)(3 22 138 150 71 54 82 103)(4 21 139 149 72 53 83 102)(5 40 140 148 73 52 84 101)(6 39 121 147 74 51 85 120)(7 38 122 146 75 50 86 119)(8 37 123 145 76 49 87 118)(9 36 124 144 77 48 88 117)(10 35 125 143 78 47 89 116)(11 34 126 142 79 46 90 115)(12 33 127 141 80 45 91 114)(13 32 128 160 61 44 92 113)(14 31 129 159 62 43 93 112)(15 30 130 158 63 42 94 111)(16 29 131 157 64 41 95 110)(17 28 132 156 65 60 96 109)(18 27 133 155 66 59 97 108)(19 26 134 154 67 58 98 107)(20 25 135 153 68 57 99 106)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 49)(22 58)(23 47)(24 56)(25 45)(26 54)(27 43)(28 52)(29 41)(30 50)(31 59)(32 48)(33 57)(34 46)(35 55)(36 44)(37 53)(38 42)(39 51)(40 60)(61 77)(62 66)(63 75)(65 73)(67 71)(68 80)(70 78)(72 76)(81 89)(82 98)(83 87)(84 96)(86 94)(88 92)(91 99)(93 97)(101 156)(102 145)(103 154)(104 143)(105 152)(106 141)(107 150)(108 159)(109 148)(110 157)(111 146)(112 155)(113 144)(114 153)(115 142)(116 151)(117 160)(118 149)(119 158)(120 147)(122 130)(123 139)(124 128)(125 137)(127 135)(129 133)(132 140)(134 138)
G:=sub<Sym(160)| (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,24,136,152,69,56,100,105)(2,23,137,151,70,55,81,104)(3,22,138,150,71,54,82,103)(4,21,139,149,72,53,83,102)(5,40,140,148,73,52,84,101)(6,39,121,147,74,51,85,120)(7,38,122,146,75,50,86,119)(8,37,123,145,76,49,87,118)(9,36,124,144,77,48,88,117)(10,35,125,143,78,47,89,116)(11,34,126,142,79,46,90,115)(12,33,127,141,80,45,91,114)(13,32,128,160,61,44,92,113)(14,31,129,159,62,43,93,112)(15,30,130,158,63,42,94,111)(16,29,131,157,64,41,95,110)(17,28,132,156,65,60,96,109)(18,27,133,155,66,59,97,108)(19,26,134,154,67,58,98,107)(20,25,135,153,68,57,99,106), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,49)(22,58)(23,47)(24,56)(25,45)(26,54)(27,43)(28,52)(29,41)(30,50)(31,59)(32,48)(33,57)(34,46)(35,55)(36,44)(37,53)(38,42)(39,51)(40,60)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(81,89)(82,98)(83,87)(84,96)(86,94)(88,92)(91,99)(93,97)(101,156)(102,145)(103,154)(104,143)(105,152)(106,141)(107,150)(108,159)(109,148)(110,157)(111,146)(112,155)(113,144)(114,153)(115,142)(116,151)(117,160)(118,149)(119,158)(120,147)(122,130)(123,139)(124,128)(125,137)(127,135)(129,133)(132,140)(134,138)>;
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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,24,136,152,69,56,100,105)(2,23,137,151,70,55,81,104)(3,22,138,150,71,54,82,103)(4,21,139,149,72,53,83,102)(5,40,140,148,73,52,84,101)(6,39,121,147,74,51,85,120)(7,38,122,146,75,50,86,119)(8,37,123,145,76,49,87,118)(9,36,124,144,77,48,88,117)(10,35,125,143,78,47,89,116)(11,34,126,142,79,46,90,115)(12,33,127,141,80,45,91,114)(13,32,128,160,61,44,92,113)(14,31,129,159,62,43,93,112)(15,30,130,158,63,42,94,111)(16,29,131,157,64,41,95,110)(17,28,132,156,65,60,96,109)(18,27,133,155,66,59,97,108)(19,26,134,154,67,58,98,107)(20,25,135,153,68,57,99,106), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,49)(22,58)(23,47)(24,56)(25,45)(26,54)(27,43)(28,52)(29,41)(30,50)(31,59)(32,48)(33,57)(34,46)(35,55)(36,44)(37,53)(38,42)(39,51)(40,60)(61,77)(62,66)(63,75)(65,73)(67,71)(68,80)(70,78)(72,76)(81,89)(82,98)(83,87)(84,96)(86,94)(88,92)(91,99)(93,97)(101,156)(102,145)(103,154)(104,143)(105,152)(106,141)(107,150)(108,159)(109,148)(110,157)(111,146)(112,155)(113,144)(114,153)(115,142)(116,151)(117,160)(118,149)(119,158)(120,147)(122,130)(123,139)(124,128)(125,137)(127,135)(129,133)(132,140)(134,138) );
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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,24,136,152,69,56,100,105),(2,23,137,151,70,55,81,104),(3,22,138,150,71,54,82,103),(4,21,139,149,72,53,83,102),(5,40,140,148,73,52,84,101),(6,39,121,147,74,51,85,120),(7,38,122,146,75,50,86,119),(8,37,123,145,76,49,87,118),(9,36,124,144,77,48,88,117),(10,35,125,143,78,47,89,116),(11,34,126,142,79,46,90,115),(12,33,127,141,80,45,91,114),(13,32,128,160,61,44,92,113),(14,31,129,159,62,43,93,112),(15,30,130,158,63,42,94,111),(16,29,131,157,64,41,95,110),(17,28,132,156,65,60,96,109),(18,27,133,155,66,59,97,108),(19,26,134,154,67,58,98,107),(20,25,135,153,68,57,99,106)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,49),(22,58),(23,47),(24,56),(25,45),(26,54),(27,43),(28,52),(29,41),(30,50),(31,59),(32,48),(33,57),(34,46),(35,55),(36,44),(37,53),(38,42),(39,51),(40,60),(61,77),(62,66),(63,75),(65,73),(67,71),(68,80),(70,78),(72,76),(81,89),(82,98),(83,87),(84,96),(86,94),(88,92),(91,99),(93,97),(101,156),(102,145),(103,154),(104,143),(105,152),(106,141),(107,150),(108,159),(109,148),(110,157),(111,146),(112,155),(113,144),(114,153),(115,142),(116,151),(117,160),(118,149),(119,158),(120,147),(122,130),(123,139),(124,128),(125,137),(127,135),(129,133),(132,140),(134,138)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | M4(2) | M4(2) | D10 | D10 | C4×D5 | C8⋊D5 | D4×D5 | Q8×D5 | D5×M4(2) |
| kernel | C20⋊5M4(2) | C20⋊3C8 | C20.8Q8 | C5×C4⋊C8 | D5×C42 | C2×C8⋊D5 | C4×Dic5 | C2×C4×D5 | C4×D5 | C4×D5 | C4⋊C8 | Dic5 | C20 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C20⋊5M4(2) ►in GL4(𝔽41) generated by
| 7 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 14 | 0 | 0 |
| 40 | 33 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 32 | 0 |
| 40 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,32,0,0,0,0,9],[8,40,0,0,14,33,0,0,0,0,0,32,0,0,40,0],[40,7,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;
C20⋊5M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes_5M_4(2) % in TeX
G:=Group("C20:5M4(2)"); // GroupNames label
G:=SmallGroup(320,464);
// by ID
G=gap.SmallGroup(320,464);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^5>;
// generators/relations