direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)×Dic5, C20.36C42, C40⋊28(C2×C4), C8⋊6(C2×Dic5), C5⋊7(C4×M4(2)), C40⋊8C4⋊26C2, C4.6(C4×Dic5), (C8×Dic5)⋊29C2, (C2×C8).274D10, (C5×M4(2))⋊8C4, C4.Dic5⋊16C4, C23.52(C4×D5), C2.7(D5×M4(2)), C10.45(C2×C42), (C2×C10).28C42, (C4×Dic5).11C4, C22.6(C4×Dic5), (C2×C20).862C23, (C2×C40).232C22, C20.199(C22×C4), (C22×C4).344D10, (C2×M4(2)).18D5, (C10×M4(2)).5C2, C10.63(C2×M4(2)), C4.34(C22×Dic5), (C22×Dic5).20C4, (C22×C20).176C22, (C4×Dic5).314C22, C4.114(C2×C4×D5), C5⋊2C8⋊26(C2×C4), (C2×C4×Dic5).9C2, C2.13(C2×C4×Dic5), C22.62(C2×C4×D5), (C2×C4).156(C4×D5), (C2×C20).269(C2×C4), (C2×C4).46(C2×Dic5), (C2×C4).804(C22×D5), (C2×C4.Dic5).21C2, (C22×C10).130(C2×C4), (C2×C10).233(C22×C4), (C2×C5⋊2C8).329C22, (C2×Dic5).155(C2×C4), SmallGroup(320,744)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 334 in 142 conjugacy classes, 91 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C2×M4(2), C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×M4(2), C2×C5⋊2C8, C4.Dic5, C4×Dic5, C4×Dic5, C2×C40, C5×M4(2), C22×Dic5, C22×C20, C8×Dic5, C40⋊8C4, C2×C4.Dic5, C2×C4×Dic5, C10×M4(2), M4(2)×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, Dic5, D10, C2×C42, C2×M4(2), C4×D5, C2×Dic5, C22×D5, C4×M4(2), C4×Dic5, C2×C4×D5, C22×Dic5, D5×M4(2), C2×C4×Dic5, M4(2)×Dic5
►On 160 points
Generators in S160
(1 53 43 74 17 85 33 69)(2 54 44 75 18 86 34 70)(3 55 45 76 19 87 35 61)(4 56 46 77 20 88 36 62)(5 57 47 78 11 89 37 63)(6 58 48 79 12 90 38 64)(7 59 49 80 13 81 39 65)(8 60 50 71 14 82 40 66)(9 51 41 72 15 83 31 67)(10 52 42 73 16 84 32 68)(21 92 146 116 137 127 151 101)(22 93 147 117 138 128 152 102)(23 94 148 118 139 129 153 103)(24 95 149 119 140 130 154 104)(25 96 150 120 131 121 155 105)(26 97 141 111 132 122 156 106)(27 98 142 112 133 123 157 107)(28 99 143 113 134 124 158 108)(29 100 144 114 135 125 159 109)(30 91 145 115 136 126 160 110)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 11)(21 26)(22 27)(23 28)(24 29)(25 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)
(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 92 6 97)(2 91 7 96)(3 100 8 95)(4 99 9 94)(5 98 10 93)(11 123 16 128)(12 122 17 127)(13 121 18 126)(14 130 19 125)(15 129 20 124)(21 64 26 69)(22 63 27 68)(23 62 28 67)(24 61 29 66)(25 70 30 65)(31 103 36 108)(32 102 37 107)(33 101 38 106)(34 110 39 105)(35 109 40 104)(41 118 46 113)(42 117 47 112)(43 116 48 111)(44 115 49 120)(45 114 50 119)(51 148 56 143)(52 147 57 142)(53 146 58 141)(54 145 59 150)(55 144 60 149)(71 140 76 135)(72 139 77 134)(73 138 78 133)(74 137 79 132)(75 136 80 131)(81 155 86 160)(82 154 87 159)(83 153 88 158)(84 152 89 157)(85 151 90 156)
G:=sub<Sym(160)| (1,53,43,74,17,85,33,69)(2,54,44,75,18,86,34,70)(3,55,45,76,19,87,35,61)(4,56,46,77,20,88,36,62)(5,57,47,78,11,89,37,63)(6,58,48,79,12,90,38,64)(7,59,49,80,13,81,39,65)(8,60,50,71,14,82,40,66)(9,51,41,72,15,83,31,67)(10,52,42,73,16,84,32,68)(21,92,146,116,137,127,151,101)(22,93,147,117,138,128,152,102)(23,94,148,118,139,129,153,103)(24,95,149,119,140,130,154,104)(25,96,150,120,131,121,155,105)(26,97,141,111,132,122,156,106)(27,98,142,112,133,123,157,107)(28,99,143,113,134,124,158,108)(29,100,144,114,135,125,159,109)(30,91,145,115,136,126,160,110), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,11)(21,26)(22,27)(23,28)(24,29)(25,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,41)(37,42)(38,43)(39,44)(40,45)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(131,136)(132,137)(133,138)(134,139)(135,140)(141,146)(142,147)(143,148)(144,149)(145,150)(151,156)(152,157)(153,158)(154,159)(155,160), (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), (1,92,6,97)(2,91,7,96)(3,100,8,95)(4,99,9,94)(5,98,10,93)(11,123,16,128)(12,122,17,127)(13,121,18,126)(14,130,19,125)(15,129,20,124)(21,64,26,69)(22,63,27,68)(23,62,28,67)(24,61,29,66)(25,70,30,65)(31,103,36,108)(32,102,37,107)(33,101,38,106)(34,110,39,105)(35,109,40,104)(41,118,46,113)(42,117,47,112)(43,116,48,111)(44,115,49,120)(45,114,50,119)(51,148,56,143)(52,147,57,142)(53,146,58,141)(54,145,59,150)(55,144,60,149)(71,140,76,135)(72,139,77,134)(73,138,78,133)(74,137,79,132)(75,136,80,131)(81,155,86,160)(82,154,87,159)(83,153,88,158)(84,152,89,157)(85,151,90,156)>;
G:=Group( (1,53,43,74,17,85,33,69)(2,54,44,75,18,86,34,70)(3,55,45,76,19,87,35,61)(4,56,46,77,20,88,36,62)(5,57,47,78,11,89,37,63)(6,58,48,79,12,90,38,64)(7,59,49,80,13,81,39,65)(8,60,50,71,14,82,40,66)(9,51,41,72,15,83,31,67)(10,52,42,73,16,84,32,68)(21,92,146,116,137,127,151,101)(22,93,147,117,138,128,152,102)(23,94,148,118,139,129,153,103)(24,95,149,119,140,130,154,104)(25,96,150,120,131,121,155,105)(26,97,141,111,132,122,156,106)(27,98,142,112,133,123,157,107)(28,99,143,113,134,124,158,108)(29,100,144,114,135,125,159,109)(30,91,145,115,136,126,160,110), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,11)(21,26)(22,27)(23,28)(24,29)(25,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,41)(37,42)(38,43)(39,44)(40,45)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(131,136)(132,137)(133,138)(134,139)(135,140)(141,146)(142,147)(143,148)(144,149)(145,150)(151,156)(152,157)(153,158)(154,159)(155,160), (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), (1,92,6,97)(2,91,7,96)(3,100,8,95)(4,99,9,94)(5,98,10,93)(11,123,16,128)(12,122,17,127)(13,121,18,126)(14,130,19,125)(15,129,20,124)(21,64,26,69)(22,63,27,68)(23,62,28,67)(24,61,29,66)(25,70,30,65)(31,103,36,108)(32,102,37,107)(33,101,38,106)(34,110,39,105)(35,109,40,104)(41,118,46,113)(42,117,47,112)(43,116,48,111)(44,115,49,120)(45,114,50,119)(51,148,56,143)(52,147,57,142)(53,146,58,141)(54,145,59,150)(55,144,60,149)(71,140,76,135)(72,139,77,134)(73,138,78,133)(74,137,79,132)(75,136,80,131)(81,155,86,160)(82,154,87,159)(83,153,88,158)(84,152,89,157)(85,151,90,156) );
G=PermutationGroup([[(1,53,43,74,17,85,33,69),(2,54,44,75,18,86,34,70),(3,55,45,76,19,87,35,61),(4,56,46,77,20,88,36,62),(5,57,47,78,11,89,37,63),(6,58,48,79,12,90,38,64),(7,59,49,80,13,81,39,65),(8,60,50,71,14,82,40,66),(9,51,41,72,15,83,31,67),(10,52,42,73,16,84,32,68),(21,92,146,116,137,127,151,101),(22,93,147,117,138,128,152,102),(23,94,148,118,139,129,153,103),(24,95,149,119,140,130,154,104),(25,96,150,120,131,121,155,105),(26,97,141,111,132,122,156,106),(27,98,142,112,133,123,157,107),(28,99,143,113,134,124,158,108),(29,100,144,114,135,125,159,109),(30,91,145,115,136,126,160,110)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,11),(21,26),(22,27),(23,28),(24,29),(25,30),(31,46),(32,47),(33,48),(34,49),(35,50),(36,41),(37,42),(38,43),(39,44),(40,45),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80),(81,86),(82,87),(83,88),(84,89),(85,90),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(131,136),(132,137),(133,138),(134,139),(135,140),(141,146),(142,147),(143,148),(144,149),(145,150),(151,156),(152,157),(153,158),(154,159),(155,160)], [(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)], [(1,92,6,97),(2,91,7,96),(3,100,8,95),(4,99,9,94),(5,98,10,93),(11,123,16,128),(12,122,17,127),(13,121,18,126),(14,130,19,125),(15,129,20,124),(21,64,26,69),(22,63,27,68),(23,62,28,67),(24,61,29,66),(25,70,30,65),(31,103,36,108),(32,102,37,107),(33,101,38,106),(34,110,39,105),(35,109,40,104),(41,118,46,113),(42,117,47,112),(43,116,48,111),(44,115,49,120),(45,114,50,119),(51,148,56,143),(52,147,57,142),(53,146,58,141),(54,145,59,150),(55,144,60,149),(71,140,76,135),(72,139,77,134),(73,138,78,133),(74,137,79,132),(75,136,80,131),(81,155,86,160),(82,154,87,159),(83,153,88,158),(84,152,89,157),(85,151,90,156)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | M4(2) | D10 | Dic5 | D10 | C4×D5 | C4×D5 | D5×M4(2) |
| kernel | M4(2)×Dic5 | C8×Dic5 | C40⋊8C4 | C2×C4.Dic5 | C2×C4×Dic5 | C10×M4(2) | C4.Dic5 | C4×Dic5 | C5×M4(2) | C22×Dic5 | C2×M4(2) | Dic5 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 8 | 4 | 2 | 8 | 4 | 8 | 2 | 12 | 4 | 8 |
Matrix representation of M4(2)×Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 11 | 12 |
| 0 | 0 | 18 | 30 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 24 |
| 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 39 | 26 | 0 | 0 |
| 14 | 2 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,11,18,0,0,12,30],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,24,1],[0,1,0,0,40,35,0,0,0,0,1,0,0,0,0,1],[39,14,0,0,26,2,0,0,0,0,40,0,0,0,0,40] >;
M4(2)×Dic5 in GAP, Magma, Sage, TeX
M_4(2)\times {\rm Dic}_5 % in TeX
G:=Group("M4(2)xDic5"); // GroupNames label
G:=SmallGroup(320,744);
// by ID
G=gap.SmallGroup(320,744);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations