metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6(C4×D5), C40⋊21(C2×C4), C40⋊C2⋊7C4, C40⋊8C4⋊7C2, C2.D8⋊12D5, (C2×C8).67D10, C10.84(C4×D4), C4⋊C4.172D10, D20.23(C2×C4), C22.92(D4×D5), Dic10⋊16(C2×C4), Dic5⋊3Q8⋊7C2, D20⋊6C4.7C2, D20⋊8C4.7C2, C20.44(C4○D4), C2.6(D8⋊D5), C10.Q16⋊23C2, C5⋊5(SD16⋊C4), C10.44(C8⋊C22), (C2×C20).305C23, C20.110(C22×C4), (C2×C40).145C22, C4.12(Q8⋊2D5), C2.6(Q16⋊D5), (C2×Dic5).226D4, (C2×D20).89C22, C2.14(D20⋊8C4), C10.73(C8.C22), (C4×Dic5).43C22, (C2×Dic10).97C22, C4.45(C2×C4×D5), (C5×C2.D8)⋊9C2, (C2×C40⋊C2).7C2, (C2×C10).310(C2×D4), (C5×C4⋊C4).98C22, (C2×C5⋊2C8).74C22, (C2×C4).408(C22×D5), SmallGroup(320,516)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊21(C2×C4)
G = < a,b,c | a40=b2=c4=1, bab=a19, cac-1=a31, bc=cb >
Subgroups: 502 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], D4 [×3], Q8 [×3], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, SD16⋊C4, C40⋊C2 [×4], C2×C5⋊2C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D20⋊6C4, C10.Q16, C40⋊8C4, C5×C2.D8, Dic5⋊3Q8, D20⋊8C4, C2×C40⋊C2, C40⋊21(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, SD16⋊C4, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D8⋊D5, Q16⋊D5, C40⋊21(C2×C4)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 65)(42 44)(43 63)(45 61)(46 80)(47 59)(48 78)(49 57)(50 76)(51 55)(52 74)(54 72)(56 70)(58 68)(60 66)(62 64)(67 79)(69 77)(71 75)(81 83)(82 102)(84 100)(85 119)(86 98)(87 117)(88 96)(89 115)(90 94)(91 113)(93 111)(95 109)(97 107)(99 105)(101 103)(104 120)(106 118)(108 116)(110 114)(122 140)(123 159)(124 138)(125 157)(126 136)(127 155)(128 134)(129 153)(130 132)(131 151)(133 149)(135 147)(137 145)(139 143)(142 160)(144 158)(146 156)(148 154)(150 152)
(1 141 92 53)(2 132 93 44)(3 123 94 75)(4 154 95 66)(5 145 96 57)(6 136 97 48)(7 127 98 79)(8 158 99 70)(9 149 100 61)(10 140 101 52)(11 131 102 43)(12 122 103 74)(13 153 104 65)(14 144 105 56)(15 135 106 47)(16 126 107 78)(17 157 108 69)(18 148 109 60)(19 139 110 51)(20 130 111 42)(21 121 112 73)(22 152 113 64)(23 143 114 55)(24 134 115 46)(25 125 116 77)(26 156 117 68)(27 147 118 59)(28 138 119 50)(29 129 120 41)(30 160 81 72)(31 151 82 63)(32 142 83 54)(33 133 84 45)(34 124 85 76)(35 155 86 67)(36 146 87 58)(37 137 88 49)(38 128 89 80)(39 159 90 71)(40 150 91 62)
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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,65)(42,44)(43,63)(45,61)(46,80)(47,59)(48,78)(49,57)(50,76)(51,55)(52,74)(54,72)(56,70)(58,68)(60,66)(62,64)(67,79)(69,77)(71,75)(81,83)(82,102)(84,100)(85,119)(86,98)(87,117)(88,96)(89,115)(90,94)(91,113)(93,111)(95,109)(97,107)(99,105)(101,103)(104,120)(106,118)(108,116)(110,114)(122,140)(123,159)(124,138)(125,157)(126,136)(127,155)(128,134)(129,153)(130,132)(131,151)(133,149)(135,147)(137,145)(139,143)(142,160)(144,158)(146,156)(148,154)(150,152), (1,141,92,53)(2,132,93,44)(3,123,94,75)(4,154,95,66)(5,145,96,57)(6,136,97,48)(7,127,98,79)(8,158,99,70)(9,149,100,61)(10,140,101,52)(11,131,102,43)(12,122,103,74)(13,153,104,65)(14,144,105,56)(15,135,106,47)(16,126,107,78)(17,157,108,69)(18,148,109,60)(19,139,110,51)(20,130,111,42)(21,121,112,73)(22,152,113,64)(23,143,114,55)(24,134,115,46)(25,125,116,77)(26,156,117,68)(27,147,118,59)(28,138,119,50)(29,129,120,41)(30,160,81,72)(31,151,82,63)(32,142,83,54)(33,133,84,45)(34,124,85,76)(35,155,86,67)(36,146,87,58)(37,137,88,49)(38,128,89,80)(39,159,90,71)(40,150,91,62)>;
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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,65)(42,44)(43,63)(45,61)(46,80)(47,59)(48,78)(49,57)(50,76)(51,55)(52,74)(54,72)(56,70)(58,68)(60,66)(62,64)(67,79)(69,77)(71,75)(81,83)(82,102)(84,100)(85,119)(86,98)(87,117)(88,96)(89,115)(90,94)(91,113)(93,111)(95,109)(97,107)(99,105)(101,103)(104,120)(106,118)(108,116)(110,114)(122,140)(123,159)(124,138)(125,157)(126,136)(127,155)(128,134)(129,153)(130,132)(131,151)(133,149)(135,147)(137,145)(139,143)(142,160)(144,158)(146,156)(148,154)(150,152), (1,141,92,53)(2,132,93,44)(3,123,94,75)(4,154,95,66)(5,145,96,57)(6,136,97,48)(7,127,98,79)(8,158,99,70)(9,149,100,61)(10,140,101,52)(11,131,102,43)(12,122,103,74)(13,153,104,65)(14,144,105,56)(15,135,106,47)(16,126,107,78)(17,157,108,69)(18,148,109,60)(19,139,110,51)(20,130,111,42)(21,121,112,73)(22,152,113,64)(23,143,114,55)(24,134,115,46)(25,125,116,77)(26,156,117,68)(27,147,118,59)(28,138,119,50)(29,129,120,41)(30,160,81,72)(31,151,82,63)(32,142,83,54)(33,133,84,45)(34,124,85,76)(35,155,86,67)(36,146,87,58)(37,137,88,49)(38,128,89,80)(39,159,90,71)(40,150,91,62) );
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)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,65),(42,44),(43,63),(45,61),(46,80),(47,59),(48,78),(49,57),(50,76),(51,55),(52,74),(54,72),(56,70),(58,68),(60,66),(62,64),(67,79),(69,77),(71,75),(81,83),(82,102),(84,100),(85,119),(86,98),(87,117),(88,96),(89,115),(90,94),(91,113),(93,111),(95,109),(97,107),(99,105),(101,103),(104,120),(106,118),(108,116),(110,114),(122,140),(123,159),(124,138),(125,157),(126,136),(127,155),(128,134),(129,153),(130,132),(131,151),(133,149),(135,147),(137,145),(139,143),(142,160),(144,158),(146,156),(148,154),(150,152)], [(1,141,92,53),(2,132,93,44),(3,123,94,75),(4,154,95,66),(5,145,96,57),(6,136,97,48),(7,127,98,79),(8,158,99,70),(9,149,100,61),(10,140,101,52),(11,131,102,43),(12,122,103,74),(13,153,104,65),(14,144,105,56),(15,135,106,47),(16,126,107,78),(17,157,108,69),(18,148,109,60),(19,139,110,51),(20,130,111,42),(21,121,112,73),(22,152,113,64),(23,143,114,55),(24,134,115,46),(25,125,116,77),(26,156,117,68),(27,147,118,59),(28,138,119,50),(29,129,120,41),(30,160,81,72),(31,151,82,63),(32,142,83,54),(33,133,84,45),(34,124,85,76),(35,155,86,67),(36,146,87,58),(37,137,88,49),(38,128,89,80),(39,159,90,71),(40,150,91,62)])
50 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | C8⋊C22 | C8.C22 | Q8⋊2D5 | D4×D5 | D8⋊D5 | Q16⋊D5 |
kernel | C40⋊21(C2×C4) | D20⋊6C4 | C10.Q16 | C40⋊8C4 | C5×C2.D8 | Dic5⋊3Q8 | D20⋊8C4 | C2×C40⋊C2 | C40⋊C2 | C2×Dic5 | C2.D8 | C20 | C4⋊C4 | C2×C8 | C8 | C10 | C10 | C4 | C22 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊21(C2×C4) ►in GL6(𝔽41)
24 | 25 | 0 | 0 | 0 | 0 |
13 | 17 | 0 | 0 | 0 | 0 |
0 | 0 | 2 | 31 | 39 | 10 |
0 | 0 | 10 | 31 | 31 | 10 |
0 | 0 | 1 | 36 | 0 | 0 |
0 | 0 | 5 | 36 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
3 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 34 | 40 | 0 | 0 |
0 | 0 | 1 | 0 | 40 | 0 |
0 | 0 | 34 | 40 | 7 | 1 |
32 | 0 | 0 | 0 | 0 | 0 |
14 | 9 | 0 | 0 | 0 | 0 |
0 | 0 | 15 | 14 | 25 | 13 |
0 | 0 | 27 | 40 | 28 | 16 |
0 | 0 | 7 | 0 | 26 | 27 |
0 | 0 | 0 | 7 | 14 | 1 |
G:=sub<GL(6,GF(41))| [24,13,0,0,0,0,25,17,0,0,0,0,0,0,2,10,1,5,0,0,31,31,36,36,0,0,39,31,0,0,0,0,10,10,0,0],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,1,34,1,34,0,0,0,40,0,40,0,0,0,0,40,7,0,0,0,0,0,1],[32,14,0,0,0,0,0,9,0,0,0,0,0,0,15,27,7,0,0,0,14,40,0,7,0,0,25,28,26,14,0,0,13,16,27,1] >;
C40⋊21(C2×C4) in GAP, Magma, Sage, TeX
C_{40}\rtimes_{21}(C_2\times C_4)
% in TeX
G:=Group("C40:21(C2xC4)");
// GroupNames label
G:=SmallGroup(320,516);
// by ID
G=gap.SmallGroup(320,516);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^19,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations