metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.44D4, C10.1Q16, Dic10⋊5C4, C10.1SD16, C2.1Dic20, C22.7D20, C4.7(C4×D5), (C2×C8).2D5, (C2×C40).2C2, C20.38(C2×C4), C5⋊3(Q8⋊C4), (C2×C4).67D10, (C2×C10).12D4, C4⋊Dic5.1C2, C2.1(C40⋊C2), C4.19(C5⋊D4), (C2×C20).80C22, (C2×Dic10).1C2, C2.7(D10⋊C4), C10.16(C22⋊C4), SmallGroup(160,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.44D4
G = < a,b,c | a20=b4=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a5b-1 >
Smallest permutation representation of C20.44D4
►Regular action 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)
(1 102 32 64)(2 101 33 63)(3 120 34 62)(4 119 35 61)(5 118 36 80)(6 117 37 79)(7 116 38 78)(8 115 39 77)(9 114 40 76)(10 113 21 75)(11 112 22 74)(12 111 23 73)(13 110 24 72)(14 109 25 71)(15 108 26 70)(16 107 27 69)(17 106 28 68)(18 105 29 67)(19 104 30 66)(20 103 31 65)(41 126 93 155)(42 125 94 154)(43 124 95 153)(44 123 96 152)(45 122 97 151)(46 121 98 150)(47 140 99 149)(48 139 100 148)(49 138 81 147)(50 137 82 146)(51 136 83 145)(52 135 84 144)(53 134 85 143)(54 133 86 142)(55 132 87 141)(56 131 88 160)(57 130 89 159)(58 129 90 158)(59 128 91 157)(60 127 92 156)
(1 131 11 121)(2 130 12 140)(3 129 13 139)(4 128 14 138)(5 127 15 137)(6 126 16 136)(7 125 17 135)(8 124 18 134)(9 123 19 133)(10 122 20 132)(21 151 31 141)(22 150 32 160)(23 149 33 159)(24 148 34 158)(25 147 35 157)(26 146 36 156)(27 145 37 155)(28 144 38 154)(29 143 39 153)(30 142 40 152)(41 102 51 112)(42 101 52 111)(43 120 53 110)(44 119 54 109)(45 118 55 108)(46 117 56 107)(47 116 57 106)(48 115 58 105)(49 114 59 104)(50 113 60 103)(61 86 71 96)(62 85 72 95)(63 84 73 94)(64 83 74 93)(65 82 75 92)(66 81 76 91)(67 100 77 90)(68 99 78 89)(69 98 79 88)(70 97 80 87)
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), (1,102,32,64)(2,101,33,63)(3,120,34,62)(4,119,35,61)(5,118,36,80)(6,117,37,79)(7,116,38,78)(8,115,39,77)(9,114,40,76)(10,113,21,75)(11,112,22,74)(12,111,23,73)(13,110,24,72)(14,109,25,71)(15,108,26,70)(16,107,27,69)(17,106,28,68)(18,105,29,67)(19,104,30,66)(20,103,31,65)(41,126,93,155)(42,125,94,154)(43,124,95,153)(44,123,96,152)(45,122,97,151)(46,121,98,150)(47,140,99,149)(48,139,100,148)(49,138,81,147)(50,137,82,146)(51,136,83,145)(52,135,84,144)(53,134,85,143)(54,133,86,142)(55,132,87,141)(56,131,88,160)(57,130,89,159)(58,129,90,158)(59,128,91,157)(60,127,92,156), (1,131,11,121)(2,130,12,140)(3,129,13,139)(4,128,14,138)(5,127,15,137)(6,126,16,136)(7,125,17,135)(8,124,18,134)(9,123,19,133)(10,122,20,132)(21,151,31,141)(22,150,32,160)(23,149,33,159)(24,148,34,158)(25,147,35,157)(26,146,36,156)(27,145,37,155)(28,144,38,154)(29,143,39,153)(30,142,40,152)(41,102,51,112)(42,101,52,111)(43,120,53,110)(44,119,54,109)(45,118,55,108)(46,117,56,107)(47,116,57,106)(48,115,58,105)(49,114,59,104)(50,113,60,103)(61,86,71,96)(62,85,72,95)(63,84,73,94)(64,83,74,93)(65,82,75,92)(66,81,76,91)(67,100,77,90)(68,99,78,89)(69,98,79,88)(70,97,80,87)>;
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), (1,102,32,64)(2,101,33,63)(3,120,34,62)(4,119,35,61)(5,118,36,80)(6,117,37,79)(7,116,38,78)(8,115,39,77)(9,114,40,76)(10,113,21,75)(11,112,22,74)(12,111,23,73)(13,110,24,72)(14,109,25,71)(15,108,26,70)(16,107,27,69)(17,106,28,68)(18,105,29,67)(19,104,30,66)(20,103,31,65)(41,126,93,155)(42,125,94,154)(43,124,95,153)(44,123,96,152)(45,122,97,151)(46,121,98,150)(47,140,99,149)(48,139,100,148)(49,138,81,147)(50,137,82,146)(51,136,83,145)(52,135,84,144)(53,134,85,143)(54,133,86,142)(55,132,87,141)(56,131,88,160)(57,130,89,159)(58,129,90,158)(59,128,91,157)(60,127,92,156), (1,131,11,121)(2,130,12,140)(3,129,13,139)(4,128,14,138)(5,127,15,137)(6,126,16,136)(7,125,17,135)(8,124,18,134)(9,123,19,133)(10,122,20,132)(21,151,31,141)(22,150,32,160)(23,149,33,159)(24,148,34,158)(25,147,35,157)(26,146,36,156)(27,145,37,155)(28,144,38,154)(29,143,39,153)(30,142,40,152)(41,102,51,112)(42,101,52,111)(43,120,53,110)(44,119,54,109)(45,118,55,108)(46,117,56,107)(47,116,57,106)(48,115,58,105)(49,114,59,104)(50,113,60,103)(61,86,71,96)(62,85,72,95)(63,84,73,94)(64,83,74,93)(65,82,75,92)(66,81,76,91)(67,100,77,90)(68,99,78,89)(69,98,79,88)(70,97,80,87) );
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)], [(1,102,32,64),(2,101,33,63),(3,120,34,62),(4,119,35,61),(5,118,36,80),(6,117,37,79),(7,116,38,78),(8,115,39,77),(9,114,40,76),(10,113,21,75),(11,112,22,74),(12,111,23,73),(13,110,24,72),(14,109,25,71),(15,108,26,70),(16,107,27,69),(17,106,28,68),(18,105,29,67),(19,104,30,66),(20,103,31,65),(41,126,93,155),(42,125,94,154),(43,124,95,153),(44,123,96,152),(45,122,97,151),(46,121,98,150),(47,140,99,149),(48,139,100,148),(49,138,81,147),(50,137,82,146),(51,136,83,145),(52,135,84,144),(53,134,85,143),(54,133,86,142),(55,132,87,141),(56,131,88,160),(57,130,89,159),(58,129,90,158),(59,128,91,157),(60,127,92,156)], [(1,131,11,121),(2,130,12,140),(3,129,13,139),(4,128,14,138),(5,127,15,137),(6,126,16,136),(7,125,17,135),(8,124,18,134),(9,123,19,133),(10,122,20,132),(21,151,31,141),(22,150,32,160),(23,149,33,159),(24,148,34,158),(25,147,35,157),(26,146,36,156),(27,145,37,155),(28,144,38,154),(29,143,39,153),(30,142,40,152),(41,102,51,112),(42,101,52,111),(43,120,53,110),(44,119,54,109),(45,118,55,108),(46,117,56,107),(47,116,57,106),(48,115,58,105),(49,114,59,104),(50,113,60,103),(61,86,71,96),(62,85,72,95),(63,84,73,94),(64,83,74,93),(65,82,75,92),(66,81,76,91),(67,100,77,90),(68,99,78,89),(69,98,79,88),(70,97,80,87)]])
C20.44D4 is a maximal subgroup of
C20.14Q16 C4×C40⋊C2 C42.264D10 C4×Dic20 C42.14D10 C42.16D10 C42.20D10 Dic20⋊9C4 C23.34D20 C23.35D20 C23.10D20 D20.32D4 D20⋊14D4 Dic10⋊14D4 C22⋊Dic20 D4.D5⋊5C4 Dic5⋊6SD16 C20⋊Q8⋊C2 (C8×Dic5)⋊C2 D4⋊(C4×D5) D4⋊2D5⋊C4 D10⋊SD16 C5⋊2C8⋊D4 C5⋊Q16⋊5C4 Dic5⋊4Q16 Dic5.3Q16 C40⋊8C4.C2 D5×Q8⋊C4 (Q8×D5)⋊C4 D10⋊Q16 C5⋊2C8.D4 Dic10.3Q8 C20⋊SD16 D20.19D4 C42.36D10 C4⋊Dic20 C20.7Q16 Dic10⋊4Q8 Dic10⋊Q8 Dic10.Q8 D10.12SD16 C20.(C4○D4) Dic10⋊2Q8 Dic10.2Q8 D10.8Q16 C2.D8⋊7D5 C23.23D20 C40⋊30D4 C40.82D4 C23.46D20 C23.49D20 C40⋊2D4 C40.4D4 (C2×D8).D5 Dic10⋊D4 Dic5⋊3SD16 (C5×Q8).D4 D10⋊8SD16 Dic10.16D4 Dic5⋊3Q16 D10⋊5Q16 C6.Dic20 Dic30⋊12C4 Dic30⋊8C4
C20.44D4 is a maximal quotient of
Dic10⋊3C8 C23.30D20 C4.Dic20 C20.47D8 C20.39C42 C6.Dic20 Dic30⋊12C4 Dic30⋊8C4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | SD16 | Q16 | D10 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | Dic20 |
| kernel | C20.44D4 | C4⋊Dic5 | C2×C40 | C2×Dic10 | Dic10 | C20 | C2×C10 | C2×C8 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C20.44D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 6 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 23 | 32 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 38 | 23 | 0 | 0 |
| 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 0 | 39 | 39 |
| 0 | 0 | 0 | 23 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 38 | 23 | 0 | 0 |
| 0 | 5 | 3 | 0 | 0 |
| 0 | 0 | 0 | 35 | 35 |
| 0 | 0 | 0 | 13 | 6 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,1,0,0,0,40,0,0,0,0,0,0,9,23,0,0,0,0,32],[32,0,0,0,0,0,38,5,0,0,0,23,3,0,0,0,0,0,39,23,0,0,0,39,2],[1,0,0,0,0,0,38,5,0,0,0,23,3,0,0,0,0,0,35,13,0,0,0,35,6] >;
C20.44D4 in GAP, Magma, Sage, TeX
C_{20}._{44}D_4 % in TeX
G:=Group("C20.44D4"); // GroupNames label
G:=SmallGroup(160,23);
// by ID
G=gap.SmallGroup(160,23);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,73,79,362,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;
// generators/relations
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