metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.104D10, C10.142+ 1+4, (C4×D4)⋊6D5, (D4×C20)⋊7C2, C20⋊7D4⋊6C2, D10⋊D4⋊6C2, C4⋊C4.279D10, (C2×D4).208D10, C42⋊2D5⋊15C2, C42⋊D5⋊31C2, (C2×C10).84C24, Dic5⋊D4⋊25C2, Dic5⋊4D4⋊44C2, D10.13D4⋊6C2, (C4×C20).237C22, (C2×C20).620C23, C22⋊C4.129D10, Dic5.Q8⋊6C2, (C2×D20).28C22, C22.1(C4○D20), C23.D10⋊5C2, (C22×C4).203D10, C4⋊Dic5.38C22, C2.17(D4⋊6D10), Dic5.36(C4○D4), (D4×C10).302C22, (C22×C20).78C22, (C2×Dic5).34C23, (C22×D5).29C23, C22.112(C23×D5), C23.164(C22×D5), D10⋊C4.65C22, C23.23D10⋊15C2, (C22×C10).154C23, C5⋊3(C22.47C24), (C4×Dic5).221C22, C10.D4.64C22, C23.D5.102C22, (C22×Dic5).92C22, (C4×C5⋊D4)⋊38C2, C2.19(D5×C4○D4), C10.36(C2×C4○D4), C2.40(C2×C4○D20), (C2×C4×D5).249C22, (C2×C10).14(C4○D4), (C2×C10.D4)⋊25C2, (C5×C4⋊C4).320C22, (C2×C4).652(C22×D5), (C2×C5⋊D4).13C22, (C5×C22⋊C4).141C22, SmallGroup(320,1212)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.104D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 838 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.47C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C42⋊D5, C42⋊2D5, C23.D10, Dic5⋊4D4, D10⋊D4, Dic5.Q8, D10.13D4, C2×C10.D4, C4×C5⋊D4, C23.23D10, C20⋊7D4, Dic5⋊D4, D4×C20, C42.104D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C4○D20, C23×D5, C2×C4○D20, D4⋊6D10, D5×C4○D4, C42.104D10
►On 160 points
Generators in S160
(1 115 95 32)(2 116 96 33)(3 117 97 34)(4 118 98 35)(5 119 99 36)(6 120 100 37)(7 111 91 38)(8 112 92 39)(9 113 93 40)(10 114 94 31)(11 104 41 21)(12 105 42 22)(13 106 43 23)(14 107 44 24)(15 108 45 25)(16 109 46 26)(17 110 47 27)(18 101 48 28)(19 102 49 29)(20 103 50 30)(51 144 124 71)(52 145 125 72)(53 146 126 73)(54 147 127 74)(55 148 128 75)(56 149 129 76)(57 150 130 77)(58 141 121 78)(59 142 122 79)(60 143 123 80)(61 154 134 87)(62 155 135 88)(63 156 136 89)(64 157 137 90)(65 158 138 81)(66 159 139 82)(67 160 140 83)(68 151 131 84)(69 152 132 85)(70 153 133 86)
(1 133 123 22)(2 106 124 61)(3 135 125 24)(4 108 126 63)(5 137 127 26)(6 110 128 65)(7 139 129 28)(8 102 130 67)(9 131 121 30)(10 104 122 69)(11 142 85 31)(12 115 86 80)(13 144 87 33)(14 117 88 72)(15 146 89 35)(16 119 90 74)(17 148 81 37)(18 111 82 76)(19 150 83 39)(20 113 84 78)(21 59 132 94)(23 51 134 96)(25 53 136 98)(27 55 138 100)(29 57 140 92)(32 153 143 42)(34 155 145 44)(36 157 147 46)(38 159 149 48)(40 151 141 50)(41 79 152 114)(43 71 154 116)(45 73 156 118)(47 75 158 120)(49 77 160 112)(52 107 97 62)(54 109 99 64)(56 101 91 66)(58 103 93 68)(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 21 123 132)(2 30 124 131)(3 29 125 140)(4 28 126 139)(5 27 127 138)(6 26 128 137)(7 25 129 136)(8 24 130 135)(9 23 121 134)(10 22 122 133)(11 115 85 80)(12 114 86 79)(13 113 87 78)(14 112 88 77)(15 111 89 76)(16 120 90 75)(17 119 81 74)(18 118 82 73)(19 117 83 72)(20 116 84 71)(31 153 142 42)(32 152 143 41)(33 151 144 50)(34 160 145 49)(35 159 146 48)(36 158 147 47)(37 157 148 46)(38 156 149 45)(39 155 150 44)(40 154 141 43)(51 68 96 103)(52 67 97 102)(53 66 98 101)(54 65 99 110)(55 64 100 109)(56 63 91 108)(57 62 92 107)(58 61 93 106)(59 70 94 105)(60 69 95 104)
G:=sub<Sym(160)| (1,115,95,32)(2,116,96,33)(3,117,97,34)(4,118,98,35)(5,119,99,36)(6,120,100,37)(7,111,91,38)(8,112,92,39)(9,113,93,40)(10,114,94,31)(11,104,41,21)(12,105,42,22)(13,106,43,23)(14,107,44,24)(15,108,45,25)(16,109,46,26)(17,110,47,27)(18,101,48,28)(19,102,49,29)(20,103,50,30)(51,144,124,71)(52,145,125,72)(53,146,126,73)(54,147,127,74)(55,148,128,75)(56,149,129,76)(57,150,130,77)(58,141,121,78)(59,142,122,79)(60,143,123,80)(61,154,134,87)(62,155,135,88)(63,156,136,89)(64,157,137,90)(65,158,138,81)(66,159,139,82)(67,160,140,83)(68,151,131,84)(69,152,132,85)(70,153,133,86), (1,133,123,22)(2,106,124,61)(3,135,125,24)(4,108,126,63)(5,137,127,26)(6,110,128,65)(7,139,129,28)(8,102,130,67)(9,131,121,30)(10,104,122,69)(11,142,85,31)(12,115,86,80)(13,144,87,33)(14,117,88,72)(15,146,89,35)(16,119,90,74)(17,148,81,37)(18,111,82,76)(19,150,83,39)(20,113,84,78)(21,59,132,94)(23,51,134,96)(25,53,136,98)(27,55,138,100)(29,57,140,92)(32,153,143,42)(34,155,145,44)(36,157,147,46)(38,159,149,48)(40,151,141,50)(41,79,152,114)(43,71,154,116)(45,73,156,118)(47,75,158,120)(49,77,160,112)(52,107,97,62)(54,109,99,64)(56,101,91,66)(58,103,93,68)(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), (1,21,123,132)(2,30,124,131)(3,29,125,140)(4,28,126,139)(5,27,127,138)(6,26,128,137)(7,25,129,136)(8,24,130,135)(9,23,121,134)(10,22,122,133)(11,115,85,80)(12,114,86,79)(13,113,87,78)(14,112,88,77)(15,111,89,76)(16,120,90,75)(17,119,81,74)(18,118,82,73)(19,117,83,72)(20,116,84,71)(31,153,142,42)(32,152,143,41)(33,151,144,50)(34,160,145,49)(35,159,146,48)(36,158,147,47)(37,157,148,46)(38,156,149,45)(39,155,150,44)(40,154,141,43)(51,68,96,103)(52,67,97,102)(53,66,98,101)(54,65,99,110)(55,64,100,109)(56,63,91,108)(57,62,92,107)(58,61,93,106)(59,70,94,105)(60,69,95,104)>;
G:=Group( (1,115,95,32)(2,116,96,33)(3,117,97,34)(4,118,98,35)(5,119,99,36)(6,120,100,37)(7,111,91,38)(8,112,92,39)(9,113,93,40)(10,114,94,31)(11,104,41,21)(12,105,42,22)(13,106,43,23)(14,107,44,24)(15,108,45,25)(16,109,46,26)(17,110,47,27)(18,101,48,28)(19,102,49,29)(20,103,50,30)(51,144,124,71)(52,145,125,72)(53,146,126,73)(54,147,127,74)(55,148,128,75)(56,149,129,76)(57,150,130,77)(58,141,121,78)(59,142,122,79)(60,143,123,80)(61,154,134,87)(62,155,135,88)(63,156,136,89)(64,157,137,90)(65,158,138,81)(66,159,139,82)(67,160,140,83)(68,151,131,84)(69,152,132,85)(70,153,133,86), (1,133,123,22)(2,106,124,61)(3,135,125,24)(4,108,126,63)(5,137,127,26)(6,110,128,65)(7,139,129,28)(8,102,130,67)(9,131,121,30)(10,104,122,69)(11,142,85,31)(12,115,86,80)(13,144,87,33)(14,117,88,72)(15,146,89,35)(16,119,90,74)(17,148,81,37)(18,111,82,76)(19,150,83,39)(20,113,84,78)(21,59,132,94)(23,51,134,96)(25,53,136,98)(27,55,138,100)(29,57,140,92)(32,153,143,42)(34,155,145,44)(36,157,147,46)(38,159,149,48)(40,151,141,50)(41,79,152,114)(43,71,154,116)(45,73,156,118)(47,75,158,120)(49,77,160,112)(52,107,97,62)(54,109,99,64)(56,101,91,66)(58,103,93,68)(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), (1,21,123,132)(2,30,124,131)(3,29,125,140)(4,28,126,139)(5,27,127,138)(6,26,128,137)(7,25,129,136)(8,24,130,135)(9,23,121,134)(10,22,122,133)(11,115,85,80)(12,114,86,79)(13,113,87,78)(14,112,88,77)(15,111,89,76)(16,120,90,75)(17,119,81,74)(18,118,82,73)(19,117,83,72)(20,116,84,71)(31,153,142,42)(32,152,143,41)(33,151,144,50)(34,160,145,49)(35,159,146,48)(36,158,147,47)(37,157,148,46)(38,156,149,45)(39,155,150,44)(40,154,141,43)(51,68,96,103)(52,67,97,102)(53,66,98,101)(54,65,99,110)(55,64,100,109)(56,63,91,108)(57,62,92,107)(58,61,93,106)(59,70,94,105)(60,69,95,104) );
G=PermutationGroup([[(1,115,95,32),(2,116,96,33),(3,117,97,34),(4,118,98,35),(5,119,99,36),(6,120,100,37),(7,111,91,38),(8,112,92,39),(9,113,93,40),(10,114,94,31),(11,104,41,21),(12,105,42,22),(13,106,43,23),(14,107,44,24),(15,108,45,25),(16,109,46,26),(17,110,47,27),(18,101,48,28),(19,102,49,29),(20,103,50,30),(51,144,124,71),(52,145,125,72),(53,146,126,73),(54,147,127,74),(55,148,128,75),(56,149,129,76),(57,150,130,77),(58,141,121,78),(59,142,122,79),(60,143,123,80),(61,154,134,87),(62,155,135,88),(63,156,136,89),(64,157,137,90),(65,158,138,81),(66,159,139,82),(67,160,140,83),(68,151,131,84),(69,152,132,85),(70,153,133,86)], [(1,133,123,22),(2,106,124,61),(3,135,125,24),(4,108,126,63),(5,137,127,26),(6,110,128,65),(7,139,129,28),(8,102,130,67),(9,131,121,30),(10,104,122,69),(11,142,85,31),(12,115,86,80),(13,144,87,33),(14,117,88,72),(15,146,89,35),(16,119,90,74),(17,148,81,37),(18,111,82,76),(19,150,83,39),(20,113,84,78),(21,59,132,94),(23,51,134,96),(25,53,136,98),(27,55,138,100),(29,57,140,92),(32,153,143,42),(34,155,145,44),(36,157,147,46),(38,159,149,48),(40,151,141,50),(41,79,152,114),(43,71,154,116),(45,73,156,118),(47,75,158,120),(49,77,160,112),(52,107,97,62),(54,109,99,64),(56,101,91,66),(58,103,93,68),(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)], [(1,21,123,132),(2,30,124,131),(3,29,125,140),(4,28,126,139),(5,27,127,138),(6,26,128,137),(7,25,129,136),(8,24,130,135),(9,23,121,134),(10,22,122,133),(11,115,85,80),(12,114,86,79),(13,113,87,78),(14,112,88,77),(15,111,89,76),(16,120,90,75),(17,119,81,74),(18,118,82,73),(19,117,83,72),(20,116,84,71),(31,153,142,42),(32,152,143,41),(33,151,144,50),(34,160,145,49),(35,159,146,48),(36,158,147,47),(37,157,148,46),(38,156,149,45),(39,155,150,44),(40,154,141,43),(51,68,96,103),(52,67,97,102),(53,66,98,101),(54,65,99,110),(55,64,100,109),(56,63,91,108),(57,62,92,107),(58,61,93,106),(59,70,94,105),(60,69,95,104)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 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 | 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 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | C42.104D10 | C42⋊D5 | C42⋊2D5 | C23.D10 | Dic5⋊4D4 | D10⋊D4 | Dic5.Q8 | D10.13D4 | C2×C10.D4 | C4×C5⋊D4 | C23.23D10 | C20⋊7D4 | Dic5⋊D4 | D4×C20 | C4×D4 | Dic5 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.104D10 ►in GL4(𝔽41) generated by
| 30 | 9 | 0 | 0 |
| 32 | 11 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 20 |
| 0 | 0 | 0 | 1 |
| 1 | 34 | 0 | 0 |
| 7 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 37 | 40 |
| 19 | 9 | 0 | 0 |
| 19 | 22 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 4 | 1 |
G:=sub<GL(4,GF(41))| [30,32,0,0,9,11,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,20,1],[1,7,0,0,34,34,0,0,0,0,1,37,0,0,0,40],[19,19,0,0,9,22,0,0,0,0,40,4,0,0,0,1] >;
C42.104D10 in GAP, Magma, Sage, TeX
C_4^2._{104}D_{10} % in TeX
G:=Group("C4^2.104D10"); // GroupNames label
G:=SmallGroup(320,1212);
// by ID
G=gap.SmallGroup(320,1212);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,794,136,12550]);
// 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,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations