metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.106D10, C10.572- 1+4, (C4×D4).13D5, C4⋊C4.314D10, C20⋊2Q8⋊22C2, (D4×C20).14C2, (C4×Dic10)⋊28C2, (C2×D4).210D10, C4.15(C4○D20), (C2×C10).86C24, Dic5⋊3Q8⋊14C2, C20.109(C4○D4), C20.48D4⋊19C2, (C2×C20).155C23, (C4×C20).148C22, C22⋊C4.107D10, C20.17D4.9C2, C23.D10⋊7C2, (C22×C4).205D10, C4.116(D4⋊2D5), C23.92(C22×D5), (D4×C10).250C22, C23.21D10⋊7C2, C4⋊Dic5.297C22, (C4×Dic5).81C22, (C2×Dic5).36C23, C22.114(C23×D5), (C22×C20).105C22, (C22×C10).156C23, C5⋊2(C22.50C24), C2.15(D4.10D10), C23.D5.103C22, (C2×Dic10).245C22, C10.D4.109C22, C2.42(C2×C4○D20), C10.38(C2×C4○D4), C2.20(C2×D4⋊2D5), (C5×C4⋊C4).322C22, (C2×C4).281(C22×D5), (C5×C22⋊C4).120C22, SmallGroup(320,1214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.106D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c9 >
Subgroups: 598 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C22.50C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×C20, D4×C10, C4×Dic10, C20⋊2Q8, C23.D10, Dic5⋊3Q8, C20.48D4, C23.21D10, C20.17D4, D4×C20, C42.106D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, D4.10D10, C42.106D10
►On 160 points
Generators in S160
(1 51 150 68)(2 69 151 52)(3 53 152 70)(4 71 153 54)(5 55 154 72)(6 73 155 56)(7 57 156 74)(8 75 157 58)(9 59 158 76)(10 77 159 60)(11 41 160 78)(12 79 141 42)(13 43 142 80)(14 61 143 44)(15 45 144 62)(16 63 145 46)(17 47 146 64)(18 65 147 48)(19 49 148 66)(20 67 149 50)(21 107 100 136)(22 137 81 108)(23 109 82 138)(24 139 83 110)(25 111 84 140)(26 121 85 112)(27 113 86 122)(28 123 87 114)(29 115 88 124)(30 125 89 116)(31 117 90 126)(32 127 91 118)(33 119 92 128)(34 129 93 120)(35 101 94 130)(36 131 95 102)(37 103 96 132)(38 133 97 104)(39 105 98 134)(40 135 99 106)
(1 46 11 56)(2 47 12 57)(3 48 13 58)(4 49 14 59)(5 50 15 60)(6 51 16 41)(7 52 17 42)(8 53 18 43)(9 54 19 44)(10 55 20 45)(21 112 31 102)(22 113 32 103)(23 114 33 104)(24 115 34 105)(25 116 35 106)(26 117 36 107)(27 118 37 108)(28 119 38 109)(29 120 39 110)(30 101 40 111)(61 158 71 148)(62 159 72 149)(63 160 73 150)(64 141 74 151)(65 142 75 152)(66 143 76 153)(67 144 77 154)(68 145 78 155)(69 146 79 156)(70 147 80 157)(81 122 91 132)(82 123 92 133)(83 124 93 134)(84 125 94 135)(85 126 95 136)(86 127 96 137)(87 128 97 138)(88 129 98 139)(89 130 99 140)(90 131 100 121)
(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)
(1 130 150 101)(2 139 151 110)(3 128 152 119)(4 137 153 108)(5 126 154 117)(6 135 155 106)(7 124 156 115)(8 133 157 104)(9 122 158 113)(10 131 159 102)(11 140 160 111)(12 129 141 120)(13 138 142 109)(14 127 143 118)(15 136 144 107)(16 125 145 116)(17 134 146 105)(18 123 147 114)(19 132 148 103)(20 121 149 112)(21 45 100 62)(22 54 81 71)(23 43 82 80)(24 52 83 69)(25 41 84 78)(26 50 85 67)(27 59 86 76)(28 48 87 65)(29 57 88 74)(30 46 89 63)(31 55 90 72)(32 44 91 61)(33 53 92 70)(34 42 93 79)(35 51 94 68)(36 60 95 77)(37 49 96 66)(38 58 97 75)(39 47 98 64)(40 56 99 73)
G:=sub<Sym(160)| (1,51,150,68)(2,69,151,52)(3,53,152,70)(4,71,153,54)(5,55,154,72)(6,73,155,56)(7,57,156,74)(8,75,157,58)(9,59,158,76)(10,77,159,60)(11,41,160,78)(12,79,141,42)(13,43,142,80)(14,61,143,44)(15,45,144,62)(16,63,145,46)(17,47,146,64)(18,65,147,48)(19,49,148,66)(20,67,149,50)(21,107,100,136)(22,137,81,108)(23,109,82,138)(24,139,83,110)(25,111,84,140)(26,121,85,112)(27,113,86,122)(28,123,87,114)(29,115,88,124)(30,125,89,116)(31,117,90,126)(32,127,91,118)(33,119,92,128)(34,129,93,120)(35,101,94,130)(36,131,95,102)(37,103,96,132)(38,133,97,104)(39,105,98,134)(40,135,99,106), (1,46,11,56)(2,47,12,57)(3,48,13,58)(4,49,14,59)(5,50,15,60)(6,51,16,41)(7,52,17,42)(8,53,18,43)(9,54,19,44)(10,55,20,45)(21,112,31,102)(22,113,32,103)(23,114,33,104)(24,115,34,105)(25,116,35,106)(26,117,36,107)(27,118,37,108)(28,119,38,109)(29,120,39,110)(30,101,40,111)(61,158,71,148)(62,159,72,149)(63,160,73,150)(64,141,74,151)(65,142,75,152)(66,143,76,153)(67,144,77,154)(68,145,78,155)(69,146,79,156)(70,147,80,157)(81,122,91,132)(82,123,92,133)(83,124,93,134)(84,125,94,135)(85,126,95,136)(86,127,96,137)(87,128,97,138)(88,129,98,139)(89,130,99,140)(90,131,100,121), (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), (1,130,150,101)(2,139,151,110)(3,128,152,119)(4,137,153,108)(5,126,154,117)(6,135,155,106)(7,124,156,115)(8,133,157,104)(9,122,158,113)(10,131,159,102)(11,140,160,111)(12,129,141,120)(13,138,142,109)(14,127,143,118)(15,136,144,107)(16,125,145,116)(17,134,146,105)(18,123,147,114)(19,132,148,103)(20,121,149,112)(21,45,100,62)(22,54,81,71)(23,43,82,80)(24,52,83,69)(25,41,84,78)(26,50,85,67)(27,59,86,76)(28,48,87,65)(29,57,88,74)(30,46,89,63)(31,55,90,72)(32,44,91,61)(33,53,92,70)(34,42,93,79)(35,51,94,68)(36,60,95,77)(37,49,96,66)(38,58,97,75)(39,47,98,64)(40,56,99,73)>;
G:=Group( (1,51,150,68)(2,69,151,52)(3,53,152,70)(4,71,153,54)(5,55,154,72)(6,73,155,56)(7,57,156,74)(8,75,157,58)(9,59,158,76)(10,77,159,60)(11,41,160,78)(12,79,141,42)(13,43,142,80)(14,61,143,44)(15,45,144,62)(16,63,145,46)(17,47,146,64)(18,65,147,48)(19,49,148,66)(20,67,149,50)(21,107,100,136)(22,137,81,108)(23,109,82,138)(24,139,83,110)(25,111,84,140)(26,121,85,112)(27,113,86,122)(28,123,87,114)(29,115,88,124)(30,125,89,116)(31,117,90,126)(32,127,91,118)(33,119,92,128)(34,129,93,120)(35,101,94,130)(36,131,95,102)(37,103,96,132)(38,133,97,104)(39,105,98,134)(40,135,99,106), (1,46,11,56)(2,47,12,57)(3,48,13,58)(4,49,14,59)(5,50,15,60)(6,51,16,41)(7,52,17,42)(8,53,18,43)(9,54,19,44)(10,55,20,45)(21,112,31,102)(22,113,32,103)(23,114,33,104)(24,115,34,105)(25,116,35,106)(26,117,36,107)(27,118,37,108)(28,119,38,109)(29,120,39,110)(30,101,40,111)(61,158,71,148)(62,159,72,149)(63,160,73,150)(64,141,74,151)(65,142,75,152)(66,143,76,153)(67,144,77,154)(68,145,78,155)(69,146,79,156)(70,147,80,157)(81,122,91,132)(82,123,92,133)(83,124,93,134)(84,125,94,135)(85,126,95,136)(86,127,96,137)(87,128,97,138)(88,129,98,139)(89,130,99,140)(90,131,100,121), (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), (1,130,150,101)(2,139,151,110)(3,128,152,119)(4,137,153,108)(5,126,154,117)(6,135,155,106)(7,124,156,115)(8,133,157,104)(9,122,158,113)(10,131,159,102)(11,140,160,111)(12,129,141,120)(13,138,142,109)(14,127,143,118)(15,136,144,107)(16,125,145,116)(17,134,146,105)(18,123,147,114)(19,132,148,103)(20,121,149,112)(21,45,100,62)(22,54,81,71)(23,43,82,80)(24,52,83,69)(25,41,84,78)(26,50,85,67)(27,59,86,76)(28,48,87,65)(29,57,88,74)(30,46,89,63)(31,55,90,72)(32,44,91,61)(33,53,92,70)(34,42,93,79)(35,51,94,68)(36,60,95,77)(37,49,96,66)(38,58,97,75)(39,47,98,64)(40,56,99,73) );
G=PermutationGroup([[(1,51,150,68),(2,69,151,52),(3,53,152,70),(4,71,153,54),(5,55,154,72),(6,73,155,56),(7,57,156,74),(8,75,157,58),(9,59,158,76),(10,77,159,60),(11,41,160,78),(12,79,141,42),(13,43,142,80),(14,61,143,44),(15,45,144,62),(16,63,145,46),(17,47,146,64),(18,65,147,48),(19,49,148,66),(20,67,149,50),(21,107,100,136),(22,137,81,108),(23,109,82,138),(24,139,83,110),(25,111,84,140),(26,121,85,112),(27,113,86,122),(28,123,87,114),(29,115,88,124),(30,125,89,116),(31,117,90,126),(32,127,91,118),(33,119,92,128),(34,129,93,120),(35,101,94,130),(36,131,95,102),(37,103,96,132),(38,133,97,104),(39,105,98,134),(40,135,99,106)], [(1,46,11,56),(2,47,12,57),(3,48,13,58),(4,49,14,59),(5,50,15,60),(6,51,16,41),(7,52,17,42),(8,53,18,43),(9,54,19,44),(10,55,20,45),(21,112,31,102),(22,113,32,103),(23,114,33,104),(24,115,34,105),(25,116,35,106),(26,117,36,107),(27,118,37,108),(28,119,38,109),(29,120,39,110),(30,101,40,111),(61,158,71,148),(62,159,72,149),(63,160,73,150),(64,141,74,151),(65,142,75,152),(66,143,76,153),(67,144,77,154),(68,145,78,155),(69,146,79,156),(70,147,80,157),(81,122,91,132),(82,123,92,133),(83,124,93,134),(84,125,94,135),(85,126,95,136),(86,127,96,137),(87,128,97,138),(88,129,98,139),(89,130,99,140),(90,131,100,121)], [(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)], [(1,130,150,101),(2,139,151,110),(3,128,152,119),(4,137,153,108),(5,126,154,117),(6,135,155,106),(7,124,156,115),(8,133,157,104),(9,122,158,113),(10,131,159,102),(11,140,160,111),(12,129,141,120),(13,138,142,109),(14,127,143,118),(15,136,144,107),(16,125,145,116),(17,134,146,105),(18,123,147,114),(19,132,148,103),(20,121,149,112),(21,45,100,62),(22,54,81,71),(23,43,82,80),(24,52,83,69),(25,41,84,78),(26,50,85,67),(27,59,86,76),(28,48,87,65),(29,57,88,74),(30,46,89,63),(31,55,90,72),(32,44,91,61),(33,53,92,70),(34,42,93,79),(35,51,94,68),(36,60,95,77),(37,49,96,66),(38,58,97,75),(39,47,98,64),(40,56,99,73)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4S | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | D4⋊2D5 | D4.10D10 |
| kernel | C42.106D10 | C4×Dic10 | C20⋊2Q8 | C23.D10 | Dic5⋊3Q8 | C20.48D4 | C23.21D10 | C20.17D4 | D4×C20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.106D10 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 31 | 32 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 |
| 0 | 0 | 10 | 39 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 39 | 21 |
| 0 | 0 | 35 | 2 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,9,31,0,0,0,32],[0,1,0,0,1,0,0,0,0,0,21,10,0,0,0,39],[0,9,0,0,9,0,0,0,0,0,39,35,0,0,21,2] >;
C42.106D10 in GAP, Magma, Sage, TeX
C_4^2._{106}D_{10} % in TeX
G:=Group("C4^2.106D10"); // GroupNames label
G:=SmallGroup(320,1214);
// by ID
G=gap.SmallGroup(320,1214);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations