metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.277D10, (C4×D20)⋊4C2, (C2×C42)⋊12D5, (C4×Dic10)⋊4C2, C20⋊7D4.20C2, C4.D20⋊34C2, C42⋊2D5⋊21C2, C42⋊D5⋊27C2, (C2×C10).23C24, C20.6Q8⋊32C2, C20.234(C4○D4), C4.118(C4○D20), C20.48D4⋊50C2, (C2×C20).696C23, (C4×C20).316C22, (C22×C4).410D10, (C2×Dic5).7C23, (C22×D5).5C23, C22.66(C23×D5), (C2×D20).212C22, C22.22(C4○D20), C4⋊Dic5.290C22, C23.219(C22×D5), C23.D5.81C22, D10⋊C4.82C22, C23.23D10⋊32C2, (C22×C20).565C22, (C22×C10).385C23, C5⋊1(C23.36C23), (C4×Dic5).211C22, C10.D4.96C22, (C2×Dic10).233C22, (C2×C4×C20)⋊14C2, (C4×C5⋊D4)⋊32C2, C10.10(C2×C4○D4), C2.12(C2×C4○D20), (C2×C4×D5).239C22, (C2×C10).99(C4○D4), (C2×C4).651(C22×D5), (C2×C5⋊D4).94C22, SmallGroup(320,1151)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.277D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 750 in 234 conjugacy classes, 103 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.36C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, C4×Dic10, C20.6Q8, C42⋊D5, C4×D20, C4.D20, C42⋊2D5, C20.48D4, C4×C5⋊D4, C23.23D10, C20⋊7D4, C2×C4×C20, C42.277D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, C4○D20, C23×D5, C2×C4○D20, C42.277D10
►On 160 points
Generators in S160
(1 120 95 37)(2 111 96 38)(3 112 97 39)(4 113 98 40)(5 114 99 31)(6 115 100 32)(7 116 91 33)(8 117 92 34)(9 118 93 35)(10 119 94 36)(11 109 41 26)(12 110 42 27)(13 101 43 28)(14 102 44 29)(15 103 45 30)(16 104 46 21)(17 105 47 22)(18 106 48 23)(19 107 49 24)(20 108 50 25)(51 149 124 76)(52 150 125 77)(53 141 126 78)(54 142 127 79)(55 143 128 80)(56 144 129 71)(57 145 130 72)(58 146 121 73)(59 147 122 74)(60 148 123 75)(61 159 134 82)(62 160 135 83)(63 151 136 84)(64 152 137 85)(65 153 138 86)(66 154 139 87)(67 155 140 88)(68 156 131 89)(69 157 132 90)(70 158 133 81)
(1 133 123 22)(2 134 124 23)(3 135 125 24)(4 136 126 25)(5 137 127 26)(6 138 128 27)(7 139 129 28)(8 140 130 29)(9 131 121 30)(10 132 122 21)(11 114 85 79)(12 115 86 80)(13 116 87 71)(14 117 88 72)(15 118 89 73)(16 119 90 74)(17 120 81 75)(18 111 82 76)(19 112 83 77)(20 113 84 78)(31 152 142 41)(32 153 143 42)(33 154 144 43)(34 155 145 44)(35 156 146 45)(36 157 147 46)(37 158 148 47)(38 159 149 48)(39 160 150 49)(40 151 141 50)(51 106 96 61)(52 107 97 62)(53 108 98 63)(54 109 99 64)(55 110 100 65)(56 101 91 66)(57 102 92 67)(58 103 93 68)(59 104 94 69)(60 105 95 70)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 153 85 42)(12 41 86 152)(13 151 87 50)(14 49 88 160)(15 159 89 48)(16 47 90 158)(17 157 81 46)(18 45 82 156)(19 155 83 44)(20 43 84 154)(21 70 132 105)(22 104 133 69)(23 68 134 103)(24 102 135 67)(25 66 136 101)(26 110 137 65)(27 64 138 109)(28 108 139 63)(29 62 140 107)(30 106 131 61)(31 143 142 32)(33 141 144 40)(34 39 145 150)(35 149 146 38)(36 37 147 148)(51 93 96 58)(52 57 97 92)(53 91 98 56)(54 55 99 100)(59 95 94 60)(71 113 116 78)(72 77 117 112)(73 111 118 76)(74 75 119 120)(79 115 114 80)
G:=sub<Sym(160)| (1,120,95,37)(2,111,96,38)(3,112,97,39)(4,113,98,40)(5,114,99,31)(6,115,100,32)(7,116,91,33)(8,117,92,34)(9,118,93,35)(10,119,94,36)(11,109,41,26)(12,110,42,27)(13,101,43,28)(14,102,44,29)(15,103,45,30)(16,104,46,21)(17,105,47,22)(18,106,48,23)(19,107,49,24)(20,108,50,25)(51,149,124,76)(52,150,125,77)(53,141,126,78)(54,142,127,79)(55,143,128,80)(56,144,129,71)(57,145,130,72)(58,146,121,73)(59,147,122,74)(60,148,123,75)(61,159,134,82)(62,160,135,83)(63,151,136,84)(64,152,137,85)(65,153,138,86)(66,154,139,87)(67,155,140,88)(68,156,131,89)(69,157,132,90)(70,158,133,81), (1,133,123,22)(2,134,124,23)(3,135,125,24)(4,136,126,25)(5,137,127,26)(6,138,128,27)(7,139,129,28)(8,140,130,29)(9,131,121,30)(10,132,122,21)(11,114,85,79)(12,115,86,80)(13,116,87,71)(14,117,88,72)(15,118,89,73)(16,119,90,74)(17,120,81,75)(18,111,82,76)(19,112,83,77)(20,113,84,78)(31,152,142,41)(32,153,143,42)(33,154,144,43)(34,155,145,44)(35,156,146,45)(36,157,147,46)(37,158,148,47)(38,159,149,48)(39,160,150,49)(40,151,141,50)(51,106,96,61)(52,107,97,62)(53,108,98,63)(54,109,99,64)(55,110,100,65)(56,101,91,66)(57,102,92,67)(58,103,93,68)(59,104,94,69)(60,105,95,70), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,153,85,42)(12,41,86,152)(13,151,87,50)(14,49,88,160)(15,159,89,48)(16,47,90,158)(17,157,81,46)(18,45,82,156)(19,155,83,44)(20,43,84,154)(21,70,132,105)(22,104,133,69)(23,68,134,103)(24,102,135,67)(25,66,136,101)(26,110,137,65)(27,64,138,109)(28,108,139,63)(29,62,140,107)(30,106,131,61)(31,143,142,32)(33,141,144,40)(34,39,145,150)(35,149,146,38)(36,37,147,148)(51,93,96,58)(52,57,97,92)(53,91,98,56)(54,55,99,100)(59,95,94,60)(71,113,116,78)(72,77,117,112)(73,111,118,76)(74,75,119,120)(79,115,114,80)>;
G:=Group( (1,120,95,37)(2,111,96,38)(3,112,97,39)(4,113,98,40)(5,114,99,31)(6,115,100,32)(7,116,91,33)(8,117,92,34)(9,118,93,35)(10,119,94,36)(11,109,41,26)(12,110,42,27)(13,101,43,28)(14,102,44,29)(15,103,45,30)(16,104,46,21)(17,105,47,22)(18,106,48,23)(19,107,49,24)(20,108,50,25)(51,149,124,76)(52,150,125,77)(53,141,126,78)(54,142,127,79)(55,143,128,80)(56,144,129,71)(57,145,130,72)(58,146,121,73)(59,147,122,74)(60,148,123,75)(61,159,134,82)(62,160,135,83)(63,151,136,84)(64,152,137,85)(65,153,138,86)(66,154,139,87)(67,155,140,88)(68,156,131,89)(69,157,132,90)(70,158,133,81), (1,133,123,22)(2,134,124,23)(3,135,125,24)(4,136,126,25)(5,137,127,26)(6,138,128,27)(7,139,129,28)(8,140,130,29)(9,131,121,30)(10,132,122,21)(11,114,85,79)(12,115,86,80)(13,116,87,71)(14,117,88,72)(15,118,89,73)(16,119,90,74)(17,120,81,75)(18,111,82,76)(19,112,83,77)(20,113,84,78)(31,152,142,41)(32,153,143,42)(33,154,144,43)(34,155,145,44)(35,156,146,45)(36,157,147,46)(37,158,148,47)(38,159,149,48)(39,160,150,49)(40,151,141,50)(51,106,96,61)(52,107,97,62)(53,108,98,63)(54,109,99,64)(55,110,100,65)(56,101,91,66)(57,102,92,67)(58,103,93,68)(59,104,94,69)(60,105,95,70), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,153,85,42)(12,41,86,152)(13,151,87,50)(14,49,88,160)(15,159,89,48)(16,47,90,158)(17,157,81,46)(18,45,82,156)(19,155,83,44)(20,43,84,154)(21,70,132,105)(22,104,133,69)(23,68,134,103)(24,102,135,67)(25,66,136,101)(26,110,137,65)(27,64,138,109)(28,108,139,63)(29,62,140,107)(30,106,131,61)(31,143,142,32)(33,141,144,40)(34,39,145,150)(35,149,146,38)(36,37,147,148)(51,93,96,58)(52,57,97,92)(53,91,98,56)(54,55,99,100)(59,95,94,60)(71,113,116,78)(72,77,117,112)(73,111,118,76)(74,75,119,120)(79,115,114,80) );
G=PermutationGroup([[(1,120,95,37),(2,111,96,38),(3,112,97,39),(4,113,98,40),(5,114,99,31),(6,115,100,32),(7,116,91,33),(8,117,92,34),(9,118,93,35),(10,119,94,36),(11,109,41,26),(12,110,42,27),(13,101,43,28),(14,102,44,29),(15,103,45,30),(16,104,46,21),(17,105,47,22),(18,106,48,23),(19,107,49,24),(20,108,50,25),(51,149,124,76),(52,150,125,77),(53,141,126,78),(54,142,127,79),(55,143,128,80),(56,144,129,71),(57,145,130,72),(58,146,121,73),(59,147,122,74),(60,148,123,75),(61,159,134,82),(62,160,135,83),(63,151,136,84),(64,152,137,85),(65,153,138,86),(66,154,139,87),(67,155,140,88),(68,156,131,89),(69,157,132,90),(70,158,133,81)], [(1,133,123,22),(2,134,124,23),(3,135,125,24),(4,136,126,25),(5,137,127,26),(6,138,128,27),(7,139,129,28),(8,140,130,29),(9,131,121,30),(10,132,122,21),(11,114,85,79),(12,115,86,80),(13,116,87,71),(14,117,88,72),(15,118,89,73),(16,119,90,74),(17,120,81,75),(18,111,82,76),(19,112,83,77),(20,113,84,78),(31,152,142,41),(32,153,143,42),(33,154,144,43),(34,155,145,44),(35,156,146,45),(36,157,147,46),(37,158,148,47),(38,159,149,48),(39,160,150,49),(40,151,141,50),(51,106,96,61),(52,107,97,62),(53,108,98,63),(54,109,99,64),(55,110,100,65),(56,101,91,66),(57,102,92,67),(58,103,93,68),(59,104,94,69),(60,105,95,70)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,153,85,42),(12,41,86,152),(13,151,87,50),(14,49,88,160),(15,159,89,48),(16,47,90,158),(17,157,81,46),(18,45,82,156),(19,155,83,44),(20,43,84,154),(21,70,132,105),(22,104,133,69),(23,68,134,103),(24,102,135,67),(25,66,136,101),(26,110,137,65),(27,64,138,109),(28,108,139,63),(29,62,140,107),(30,106,131,61),(31,143,142,32),(33,141,144,40),(34,39,145,150),(35,149,146,38),(36,37,147,148),(51,93,96,58),(52,57,97,92),(53,91,98,56),(54,55,99,100),(59,95,94,60),(71,113,116,78),(72,77,117,112),(73,111,118,76),(74,75,119,120),(79,115,114,80)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | C4○D20 | C4○D20 |
| kernel | C42.277D10 | C4×Dic10 | C20.6Q8 | C42⋊D5 | C4×D20 | C4.D20 | C42⋊2D5 | C20.48D4 | C4×C5⋊D4 | C23.23D10 | C20⋊7D4 | C2×C4×C20 | C2×C42 | C20 | C2×C10 | C42 | C22×C4 | C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 4 | 8 | 6 | 32 | 16 |
Matrix representation of C42.277D10 ►in GL4(𝔽41) generated by
| 2 | 28 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 0 | 0 | 23 | 6 |
| 0 | 0 | 35 | 18 |
| 2 | 28 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 18 | 20 | 0 | 0 |
| 21 | 21 | 0 | 0 |
| 0 | 0 | 20 | 21 |
| 0 | 0 | 20 | 23 |
| 20 | 18 | 0 | 0 |
| 21 | 21 | 0 | 0 |
| 0 | 0 | 21 | 20 |
| 0 | 0 | 23 | 20 |
G:=sub<GL(4,GF(41))| [2,13,0,0,28,39,0,0,0,0,23,35,0,0,6,18],[2,13,0,0,28,39,0,0,0,0,32,0,0,0,0,32],[18,21,0,0,20,21,0,0,0,0,20,20,0,0,21,23],[20,21,0,0,18,21,0,0,0,0,21,23,0,0,20,20] >;
C42.277D10 in GAP, Magma, Sage, TeX
C_4^2._{277}D_{10} % in TeX
G:=Group("C4^2.277D10"); // GroupNames label
G:=SmallGroup(320,1151);
// by ID
G=gap.SmallGroup(320,1151);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations