direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4.4D8, C20.42D8, C20.32SD16, (C4×C8)⋊5C10, C4⋊Q8⋊5C10, C4.4(C5×D8), (C4×C40)⋊10C2, C2.9(C10×D8), C10.81(C2×D8), D4⋊C4⋊3C10, C4.3(C5×SD16), (C2×C20).419D4, C4⋊1D4.4C10, C42.76(C2×C10), C2.14(C10×SD16), C10.94(C2×SD16), C20.268(C4○D4), (C2×C40).366C22, (C4×C20).360C22, (C2×C20).942C23, C22.107(D4×C10), C10.71(C4.4D4), (D4×C10).198C22, (C5×C4⋊Q8)⋊26C2, C4.13(C5×C4○D4), (C2×C4).75(C5×D4), (C5×D4⋊C4)⋊3C2, C4⋊C4.17(C2×C10), (C2×C8).68(C2×C10), C2.9(C5×C4.4D4), (C2×D4).21(C2×C10), (C5×C4⋊1D4).11C2, (C2×C10).663(C2×D4), (C5×C4⋊C4).237C22, (C2×C4).117(C22×C10), SmallGroup(320,987)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.4D8
G = < a,b,c,d | a5=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 258 in 118 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C4.4D8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, D4×C10, D4×C10, Q8×C10, C4×C40, C5×D4⋊C4, C5×C4⋊1D4, C5×C4⋊Q8, C5×C4.4D8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, SD16, C2×D4, C4○D4, C2×C10, C4.4D4, C2×D8, C2×SD16, C5×D4, C22×C10, C4.4D8, C5×D8, C5×SD16, D4×C10, C5×C4○D4, C5×C4.4D4, C10×D8, C10×SD16, C5×C4.4D8
►On 160 points
Generators in S160
(1 28 103 20 95)(2 29 104 21 96)(3 30 97 22 89)(4 31 98 23 90)(5 32 99 24 91)(6 25 100 17 92)(7 26 101 18 93)(8 27 102 19 94)(9 41 116 33 108)(10 42 117 34 109)(11 43 118 35 110)(12 44 119 36 111)(13 45 120 37 112)(14 46 113 38 105)(15 47 114 39 106)(16 48 115 40 107)(49 124 140 57 132)(50 125 141 58 133)(51 126 142 59 134)(52 127 143 60 135)(53 128 144 61 136)(54 121 137 62 129)(55 122 138 63 130)(56 123 139 64 131)(65 85 156 73 148)(66 86 157 74 149)(67 87 158 75 150)(68 88 159 76 151)(69 81 160 77 152)(70 82 153 78 145)(71 83 154 79 146)(72 84 155 80 147)
(1 65 123 112)(2 66 124 105)(3 67 125 106)(4 68 126 107)(5 69 127 108)(6 70 128 109)(7 71 121 110)(8 72 122 111)(9 32 81 143)(10 25 82 144)(11 26 83 137)(12 27 84 138)(13 28 85 139)(14 29 86 140)(15 30 87 141)(16 31 88 142)(17 78 136 117)(18 79 129 118)(19 80 130 119)(20 73 131 120)(21 74 132 113)(22 75 133 114)(23 76 134 115)(24 77 135 116)(33 91 152 52)(34 92 145 53)(35 93 146 54)(36 94 147 55)(37 95 148 56)(38 96 149 49)(39 89 150 50)(40 90 151 51)(41 99 160 60)(42 100 153 61)(43 101 154 62)(44 102 155 63)(45 103 156 64)(46 104 157 57)(47 97 158 58)(48 98 159 59)
(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 72 123 111)(2 110 124 71)(3 70 125 109)(4 108 126 69)(5 68 127 107)(6 106 128 67)(7 66 121 105)(8 112 122 65)(9 142 81 31)(10 30 82 141)(11 140 83 29)(12 28 84 139)(13 138 85 27)(14 26 86 137)(15 144 87 25)(16 32 88 143)(17 114 136 75)(18 74 129 113)(19 120 130 73)(20 80 131 119)(21 118 132 79)(22 78 133 117)(23 116 134 77)(24 76 135 115)(33 51 152 90)(34 89 145 50)(35 49 146 96)(36 95 147 56)(37 55 148 94)(38 93 149 54)(39 53 150 92)(40 91 151 52)(41 59 160 98)(42 97 153 58)(43 57 154 104)(44 103 155 64)(45 63 156 102)(46 101 157 62)(47 61 158 100)(48 99 159 60)
G:=sub<Sym(160)| (1,28,103,20,95)(2,29,104,21,96)(3,30,97,22,89)(4,31,98,23,90)(5,32,99,24,91)(6,25,100,17,92)(7,26,101,18,93)(8,27,102,19,94)(9,41,116,33,108)(10,42,117,34,109)(11,43,118,35,110)(12,44,119,36,111)(13,45,120,37,112)(14,46,113,38,105)(15,47,114,39,106)(16,48,115,40,107)(49,124,140,57,132)(50,125,141,58,133)(51,126,142,59,134)(52,127,143,60,135)(53,128,144,61,136)(54,121,137,62,129)(55,122,138,63,130)(56,123,139,64,131)(65,85,156,73,148)(66,86,157,74,149)(67,87,158,75,150)(68,88,159,76,151)(69,81,160,77,152)(70,82,153,78,145)(71,83,154,79,146)(72,84,155,80,147), (1,65,123,112)(2,66,124,105)(3,67,125,106)(4,68,126,107)(5,69,127,108)(6,70,128,109)(7,71,121,110)(8,72,122,111)(9,32,81,143)(10,25,82,144)(11,26,83,137)(12,27,84,138)(13,28,85,139)(14,29,86,140)(15,30,87,141)(16,31,88,142)(17,78,136,117)(18,79,129,118)(19,80,130,119)(20,73,131,120)(21,74,132,113)(22,75,133,114)(23,76,134,115)(24,77,135,116)(33,91,152,52)(34,92,145,53)(35,93,146,54)(36,94,147,55)(37,95,148,56)(38,96,149,49)(39,89,150,50)(40,90,151,51)(41,99,160,60)(42,100,153,61)(43,101,154,62)(44,102,155,63)(45,103,156,64)(46,104,157,57)(47,97,158,58)(48,98,159,59), (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), (1,72,123,111)(2,110,124,71)(3,70,125,109)(4,108,126,69)(5,68,127,107)(6,106,128,67)(7,66,121,105)(8,112,122,65)(9,142,81,31)(10,30,82,141)(11,140,83,29)(12,28,84,139)(13,138,85,27)(14,26,86,137)(15,144,87,25)(16,32,88,143)(17,114,136,75)(18,74,129,113)(19,120,130,73)(20,80,131,119)(21,118,132,79)(22,78,133,117)(23,116,134,77)(24,76,135,115)(33,51,152,90)(34,89,145,50)(35,49,146,96)(36,95,147,56)(37,55,148,94)(38,93,149,54)(39,53,150,92)(40,91,151,52)(41,59,160,98)(42,97,153,58)(43,57,154,104)(44,103,155,64)(45,63,156,102)(46,101,157,62)(47,61,158,100)(48,99,159,60)>;
G:=Group( (1,28,103,20,95)(2,29,104,21,96)(3,30,97,22,89)(4,31,98,23,90)(5,32,99,24,91)(6,25,100,17,92)(7,26,101,18,93)(8,27,102,19,94)(9,41,116,33,108)(10,42,117,34,109)(11,43,118,35,110)(12,44,119,36,111)(13,45,120,37,112)(14,46,113,38,105)(15,47,114,39,106)(16,48,115,40,107)(49,124,140,57,132)(50,125,141,58,133)(51,126,142,59,134)(52,127,143,60,135)(53,128,144,61,136)(54,121,137,62,129)(55,122,138,63,130)(56,123,139,64,131)(65,85,156,73,148)(66,86,157,74,149)(67,87,158,75,150)(68,88,159,76,151)(69,81,160,77,152)(70,82,153,78,145)(71,83,154,79,146)(72,84,155,80,147), (1,65,123,112)(2,66,124,105)(3,67,125,106)(4,68,126,107)(5,69,127,108)(6,70,128,109)(7,71,121,110)(8,72,122,111)(9,32,81,143)(10,25,82,144)(11,26,83,137)(12,27,84,138)(13,28,85,139)(14,29,86,140)(15,30,87,141)(16,31,88,142)(17,78,136,117)(18,79,129,118)(19,80,130,119)(20,73,131,120)(21,74,132,113)(22,75,133,114)(23,76,134,115)(24,77,135,116)(33,91,152,52)(34,92,145,53)(35,93,146,54)(36,94,147,55)(37,95,148,56)(38,96,149,49)(39,89,150,50)(40,90,151,51)(41,99,160,60)(42,100,153,61)(43,101,154,62)(44,102,155,63)(45,103,156,64)(46,104,157,57)(47,97,158,58)(48,98,159,59), (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), (1,72,123,111)(2,110,124,71)(3,70,125,109)(4,108,126,69)(5,68,127,107)(6,106,128,67)(7,66,121,105)(8,112,122,65)(9,142,81,31)(10,30,82,141)(11,140,83,29)(12,28,84,139)(13,138,85,27)(14,26,86,137)(15,144,87,25)(16,32,88,143)(17,114,136,75)(18,74,129,113)(19,120,130,73)(20,80,131,119)(21,118,132,79)(22,78,133,117)(23,116,134,77)(24,76,135,115)(33,51,152,90)(34,89,145,50)(35,49,146,96)(36,95,147,56)(37,55,148,94)(38,93,149,54)(39,53,150,92)(40,91,151,52)(41,59,160,98)(42,97,153,58)(43,57,154,104)(44,103,155,64)(45,63,156,102)(46,101,157,62)(47,61,158,100)(48,99,159,60) );
G=PermutationGroup([[(1,28,103,20,95),(2,29,104,21,96),(3,30,97,22,89),(4,31,98,23,90),(5,32,99,24,91),(6,25,100,17,92),(7,26,101,18,93),(8,27,102,19,94),(9,41,116,33,108),(10,42,117,34,109),(11,43,118,35,110),(12,44,119,36,111),(13,45,120,37,112),(14,46,113,38,105),(15,47,114,39,106),(16,48,115,40,107),(49,124,140,57,132),(50,125,141,58,133),(51,126,142,59,134),(52,127,143,60,135),(53,128,144,61,136),(54,121,137,62,129),(55,122,138,63,130),(56,123,139,64,131),(65,85,156,73,148),(66,86,157,74,149),(67,87,158,75,150),(68,88,159,76,151),(69,81,160,77,152),(70,82,153,78,145),(71,83,154,79,146),(72,84,155,80,147)], [(1,65,123,112),(2,66,124,105),(3,67,125,106),(4,68,126,107),(5,69,127,108),(6,70,128,109),(7,71,121,110),(8,72,122,111),(9,32,81,143),(10,25,82,144),(11,26,83,137),(12,27,84,138),(13,28,85,139),(14,29,86,140),(15,30,87,141),(16,31,88,142),(17,78,136,117),(18,79,129,118),(19,80,130,119),(20,73,131,120),(21,74,132,113),(22,75,133,114),(23,76,134,115),(24,77,135,116),(33,91,152,52),(34,92,145,53),(35,93,146,54),(36,94,147,55),(37,95,148,56),(38,96,149,49),(39,89,150,50),(40,90,151,51),(41,99,160,60),(42,100,153,61),(43,101,154,62),(44,102,155,63),(45,103,156,64),(46,104,157,57),(47,97,158,58),(48,98,159,59)], [(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)], [(1,72,123,111),(2,110,124,71),(3,70,125,109),(4,108,126,69),(5,68,127,107),(6,106,128,67),(7,66,121,105),(8,112,122,65),(9,142,81,31),(10,30,82,141),(11,140,83,29),(12,28,84,139),(13,138,85,27),(14,26,86,137),(15,144,87,25),(16,32,88,143),(17,114,136,75),(18,74,129,113),(19,120,130,73),(20,80,131,119),(21,118,132,79),(22,78,133,117),(23,116,134,77),(24,76,135,115),(33,51,152,90),(34,89,145,50),(35,49,146,96),(36,95,147,56),(37,55,148,94),(38,93,149,54),(39,53,150,92),(40,91,151,52),(41,59,160,98),(42,97,153,58),(43,57,154,104),(44,103,155,64),(45,63,156,102),(46,101,157,62),(47,61,158,100),(48,99,159,60)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20X | 20Y | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | D8 | SD16 | C4○D4 | C5×D4 | C5×D8 | C5×SD16 | C5×C4○D4 |
| kernel | C5×C4.4D8 | C4×C40 | C5×D4⋊C4 | C5×C4⋊1D4 | C5×C4⋊Q8 | C4.4D8 | C4×C8 | D4⋊C4 | C4⋊1D4 | C4⋊Q8 | C2×C20 | C20 | C20 | C20 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C5×C4.4D8 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 40 | 2 | 0 | 0 |
| 40 | 1 | 0 | 0 |
| 0 | 0 | 11 | 16 |
| 0 | 0 | 18 | 30 |
| 11 | 30 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 11 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 0 | 17 | 22 |
| 0 | 0 | 39 | 24 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[40,40,0,0,2,1,0,0,0,0,11,18,0,0,16,30],[11,26,0,0,30,0,0,0,0,0,32,0,0,0,0,32],[0,26,0,0,11,0,0,0,0,0,17,39,0,0,22,24] >;
C5×C4.4D8 in GAP, Magma, Sage, TeX
C_5\times C_4._4D_8
% in TeX
G:=Group("C5xC4.4D8"); // GroupNames label
G:=SmallGroup(320,987);
// by ID
G=gap.SmallGroup(320,987);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,568,1766,226,7004,172,10085,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations