direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊8D4, C40⋊34D4, C8⋊8(C5×D4), C4.Q8⋊6C10, (C22×C8)⋊9C10, (C2×C10)⋊7SD16, C4.57(D4×C10), C22⋊Q8⋊3C10, D4⋊C4⋊1C10, (C22×C40)⋊23C2, Q8⋊C4⋊1C10, (C2×C20).363D4, C4⋊D4.3C10, C20.464(C2×D4), C2.9(C10×SD16), C22⋊1(C5×SD16), C23.25(C5×D4), (C10×SD16)⋊30C2, (C2×SD16)⋊13C10, C10.89(C2×SD16), C22.89(D4×C10), C10.123(C4○D8), C20.262(C4○D4), (C2×C20).924C23, (C2×C40).363C22, (C22×C10).129D4, C10.148(C4⋊D4), (D4×C10).189C22, (Q8×C10).163C22, (C22×C20).591C22, C4.7(C5×C4○D4), C4⋊C4.5(C2×C10), (C5×C4.Q8)⋊21C2, C2.10(C5×C4○D8), (C2×C4).53(C5×D4), (C5×D4⋊C4)⋊1C2, (C2×C8).93(C2×C10), (C5×Q8⋊C4)⋊1C2, C2.17(C5×C4⋊D4), (C5×C22⋊Q8)⋊30C2, (C2×Q8).7(C2×C10), (C2×D4).12(C2×C10), (C5×C4⋊D4).13C2, (C2×C10).645(C2×D4), (C5×C4⋊C4).227C22, (C2×C4).99(C22×C10), (C22×C4).120(C2×C10), SmallGroup(320,966)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊8D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >
Subgroups: 234 in 124 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C40, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C8⋊8D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×SD16, C22×C20, D4×C10, D4×C10, Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C4.Q8, C5×C4⋊D4, C5×C22⋊Q8, C22×C40, C10×SD16, C5×C8⋊8D4
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×SD16, C4○D8, C5×D4, C22×C10, C8⋊8D4, C5×SD16, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×SD16, C5×C4○D8, C5×C8⋊8D4
►On 160 points
Generators in S160
(1 111 31 103 23)(2 112 32 104 24)(3 105 25 97 17)(4 106 26 98 18)(5 107 27 99 19)(6 108 28 100 20)(7 109 29 101 21)(8 110 30 102 22)(9 90 42 114 34)(10 91 43 115 35)(11 92 44 116 36)(12 93 45 117 37)(13 94 46 118 38)(14 95 47 119 39)(15 96 48 120 40)(16 89 41 113 33)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 80 152)
(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 39 127 71)(2 34 128 66)(3 37 121 69)(4 40 122 72)(5 35 123 67)(6 38 124 70)(7 33 125 65)(8 36 126 68)(9 144 88 112)(10 139 81 107)(11 142 82 110)(12 137 83 105)(13 140 84 108)(14 143 85 111)(15 138 86 106)(16 141 87 109)(17 117 49 149)(18 120 50 152)(19 115 51 147)(20 118 52 150)(21 113 53 145)(22 116 54 148)(23 119 55 151)(24 114 56 146)(25 93 57 157)(26 96 58 160)(27 91 59 155)(28 94 60 158)(29 89 61 153)(30 92 62 156)(31 95 63 159)(32 90 64 154)(41 133 73 101)(42 136 74 104)(43 131 75 99)(44 134 76 102)(45 129 77 97)(46 132 78 100)(47 135 79 103)(48 130 80 98)
(2 4)(3 7)(6 8)(9 86)(10 81)(11 84)(12 87)(13 82)(14 85)(15 88)(16 83)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 69)(34 72)(35 67)(36 70)(37 65)(38 68)(39 71)(40 66)(41 77)(42 80)(43 75)(44 78)(45 73)(46 76)(47 79)(48 74)(49 53)(50 56)(52 54)(57 61)(58 64)(60 62)(89 157)(90 160)(91 155)(92 158)(93 153)(94 156)(95 159)(96 154)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)(113 149)(114 152)(115 147)(116 150)(117 145)(118 148)(119 151)(120 146)(121 125)(122 128)(124 126)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)
G:=sub<Sym(160)| (1,111,31,103,23)(2,112,32,104,24)(3,105,25,97,17)(4,106,26,98,18)(5,107,27,99,19)(6,108,28,100,20)(7,109,29,101,21)(8,110,30,102,22)(9,90,42,114,34)(10,91,43,115,35)(11,92,44,116,36)(12,93,45,117,37)(13,94,46,118,38)(14,95,47,119,39)(15,96,48,120,40)(16,89,41,113,33)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,87,153,73,145)(66,88,154,74,146)(67,81,155,75,147)(68,82,156,76,148)(69,83,157,77,149)(70,84,158,78,150)(71,85,159,79,151)(72,86,160,80,152), (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), (1,39,127,71)(2,34,128,66)(3,37,121,69)(4,40,122,72)(5,35,123,67)(6,38,124,70)(7,33,125,65)(8,36,126,68)(9,144,88,112)(10,139,81,107)(11,142,82,110)(12,137,83,105)(13,140,84,108)(14,143,85,111)(15,138,86,106)(16,141,87,109)(17,117,49,149)(18,120,50,152)(19,115,51,147)(20,118,52,150)(21,113,53,145)(22,116,54,148)(23,119,55,151)(24,114,56,146)(25,93,57,157)(26,96,58,160)(27,91,59,155)(28,94,60,158)(29,89,61,153)(30,92,62,156)(31,95,63,159)(32,90,64,154)(41,133,73,101)(42,136,74,104)(43,131,75,99)(44,134,76,102)(45,129,77,97)(46,132,78,100)(47,135,79,103)(48,130,80,98), (2,4)(3,7)(6,8)(9,86)(10,81)(11,84)(12,87)(13,82)(14,85)(15,88)(16,83)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,69)(34,72)(35,67)(36,70)(37,65)(38,68)(39,71)(40,66)(41,77)(42,80)(43,75)(44,78)(45,73)(46,76)(47,79)(48,74)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)(89,157)(90,160)(91,155)(92,158)(93,153)(94,156)(95,159)(96,154)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)(113,149)(114,152)(115,147)(116,150)(117,145)(118,148)(119,151)(120,146)(121,125)(122,128)(124,126)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142)>;
G:=Group( (1,111,31,103,23)(2,112,32,104,24)(3,105,25,97,17)(4,106,26,98,18)(5,107,27,99,19)(6,108,28,100,20)(7,109,29,101,21)(8,110,30,102,22)(9,90,42,114,34)(10,91,43,115,35)(11,92,44,116,36)(12,93,45,117,37)(13,94,46,118,38)(14,95,47,119,39)(15,96,48,120,40)(16,89,41,113,33)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,87,153,73,145)(66,88,154,74,146)(67,81,155,75,147)(68,82,156,76,148)(69,83,157,77,149)(70,84,158,78,150)(71,85,159,79,151)(72,86,160,80,152), (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), (1,39,127,71)(2,34,128,66)(3,37,121,69)(4,40,122,72)(5,35,123,67)(6,38,124,70)(7,33,125,65)(8,36,126,68)(9,144,88,112)(10,139,81,107)(11,142,82,110)(12,137,83,105)(13,140,84,108)(14,143,85,111)(15,138,86,106)(16,141,87,109)(17,117,49,149)(18,120,50,152)(19,115,51,147)(20,118,52,150)(21,113,53,145)(22,116,54,148)(23,119,55,151)(24,114,56,146)(25,93,57,157)(26,96,58,160)(27,91,59,155)(28,94,60,158)(29,89,61,153)(30,92,62,156)(31,95,63,159)(32,90,64,154)(41,133,73,101)(42,136,74,104)(43,131,75,99)(44,134,76,102)(45,129,77,97)(46,132,78,100)(47,135,79,103)(48,130,80,98), (2,4)(3,7)(6,8)(9,86)(10,81)(11,84)(12,87)(13,82)(14,85)(15,88)(16,83)(17,21)(18,24)(20,22)(25,29)(26,32)(28,30)(33,69)(34,72)(35,67)(36,70)(37,65)(38,68)(39,71)(40,66)(41,77)(42,80)(43,75)(44,78)(45,73)(46,76)(47,79)(48,74)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)(89,157)(90,160)(91,155)(92,158)(93,153)(94,156)(95,159)(96,154)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)(113,149)(114,152)(115,147)(116,150)(117,145)(118,148)(119,151)(120,146)(121,125)(122,128)(124,126)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142) );
G=PermutationGroup([[(1,111,31,103,23),(2,112,32,104,24),(3,105,25,97,17),(4,106,26,98,18),(5,107,27,99,19),(6,108,28,100,20),(7,109,29,101,21),(8,110,30,102,22),(9,90,42,114,34),(10,91,43,115,35),(11,92,44,116,36),(12,93,45,117,37),(13,94,46,118,38),(14,95,47,119,39),(15,96,48,120,40),(16,89,41,113,33),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,87,153,73,145),(66,88,154,74,146),(67,81,155,75,147),(68,82,156,76,148),(69,83,157,77,149),(70,84,158,78,150),(71,85,159,79,151),(72,86,160,80,152)], [(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)], [(1,39,127,71),(2,34,128,66),(3,37,121,69),(4,40,122,72),(5,35,123,67),(6,38,124,70),(7,33,125,65),(8,36,126,68),(9,144,88,112),(10,139,81,107),(11,142,82,110),(12,137,83,105),(13,140,84,108),(14,143,85,111),(15,138,86,106),(16,141,87,109),(17,117,49,149),(18,120,50,152),(19,115,51,147),(20,118,52,150),(21,113,53,145),(22,116,54,148),(23,119,55,151),(24,114,56,146),(25,93,57,157),(26,96,58,160),(27,91,59,155),(28,94,60,158),(29,89,61,153),(30,92,62,156),(31,95,63,159),(32,90,64,154),(41,133,73,101),(42,136,74,104),(43,131,75,99),(44,134,76,102),(45,129,77,97),(46,132,78,100),(47,135,79,103),(48,130,80,98)], [(2,4),(3,7),(6,8),(9,86),(10,81),(11,84),(12,87),(13,82),(14,85),(15,88),(16,83),(17,21),(18,24),(20,22),(25,29),(26,32),(28,30),(33,69),(34,72),(35,67),(36,70),(37,65),(38,68),(39,71),(40,66),(41,77),(42,80),(43,75),(44,78),(45,73),(46,76),(47,79),(48,74),(49,53),(50,56),(52,54),(57,61),(58,64),(60,62),(89,157),(90,160),(91,155),(92,158),(93,153),(94,156),(95,159),(96,154),(97,101),(98,104),(100,102),(105,109),(106,112),(108,110),(113,149),(114,152),(115,147),(116,150),(117,145),(118,148),(119,151),(120,146),(121,125),(122,128),(124,126),(129,133),(130,136),(132,134),(137,141),(138,144),(140,142)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 20A | ··· | 20P | 20Q | ··· | 20AB | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | C4○D4 | SD16 | C4○D8 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | C5×SD16 | C5×C4○D8 |
| kernel | C5×C8⋊8D4 | C5×D4⋊C4 | C5×Q8⋊C4 | C5×C4.Q8 | C5×C4⋊D4 | C5×C22⋊Q8 | C22×C40 | C10×SD16 | C8⋊8D4 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C4⋊D4 | C22⋊Q8 | C22×C8 | C2×SD16 | C40 | C2×C20 | C22×C10 | C20 | C2×C10 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C5×C8⋊8D4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 15 | 0 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,9,0,0,0,0,0,11,15,0,0,11,0],[0,1,0,0,40,0,0,0,0,0,9,0,0,0,18,32],[1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,40] >;
C5×C8⋊8D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_8D_4
% in TeX
G:=Group("C5xC8:8D4"); // GroupNames label
G:=SmallGroup(320,966);
// by ID
G=gap.SmallGroup(320,966);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
// generators/relations