metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.132D10, C10.122- (1+4), (C4×Q8)⋊14D5, (Q8×C20)⋊16C2, C4⋊C4.299D10, D10⋊3Q8⋊9C2, (C4×D20).22C2, C4.68(C4○D20), C42⋊D5⋊34C2, C42⋊2D5⋊18C2, (C2×Q8).180D10, C20.6Q8⋊19C2, D10.32(C4○D4), C20.119(C4○D4), (C4×C20).177C22, (C2×C10).125C24, (C2×C20).623C23, Dic5.Q8⋊9C2, D10.13D4.1C2, (C2×D20).226C22, C4⋊Dic5.309C22, (Q8×C10).225C22, (C2×Dic5).56C23, (C22×D5).47C23, C22.146(C23×D5), D10⋊C4.89C22, C5⋊5(C22.46C24), (C4×Dic5).229C22, C10.D4.76C22, C2.13(Q8.10D10), (D5×C4⋊C4)⋊19C2, C2.32(D5×C4○D4), C4⋊C4⋊7D5⋊17C2, C2.64(C2×C4○D20), (C2×C4×D5).84C22, C10.147(C2×C4○D4), (C5×C4⋊C4).353C22, (C2×C4).289(C22×D5), SmallGroup(320,1253)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 694 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.46C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×3], D10⋊C4 [×2], D10⋊C4 [×6], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, Q8×C10, C20.6Q8, C42⋊D5 [×2], C4×D20, C42⋊2D5 [×2], Dic5.Q8 [×2], D5×C4⋊C4, C4⋊C4⋊7D5, D10.13D4 [×2], D10⋊3Q8 [×2], Q8×C20, C42.132D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D5 [×7], C22.46C24, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D5×C4○D4, C42.132D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 143 62 134)(2 135 63 144)(3 145 64 136)(4 137 65 146)(5 147 66 138)(6 139 67 148)(7 149 68 140)(8 121 69 150)(9 151 70 122)(10 123 71 152)(11 153 72 124)(12 125 73 154)(13 155 74 126)(14 127 75 156)(15 157 76 128)(16 129 77 158)(17 159 78 130)(18 131 79 160)(19 141 80 132)(20 133 61 142)(21 110 91 46)(22 47 92 111)(23 112 93 48)(24 49 94 113)(25 114 95 50)(26 51 96 115)(27 116 97 52)(28 53 98 117)(29 118 99 54)(30 55 100 119)(31 120 81 56)(32 57 82 101)(33 102 83 58)(34 59 84 103)(35 104 85 60)(36 41 86 105)(37 106 87 42)(38 43 88 107)(39 108 89 44)(40 45 90 109)
(1 93 72 33)(2 94 73 34)(3 95 74 35)(4 96 75 36)(5 97 76 37)(6 98 77 38)(7 99 78 39)(8 100 79 40)(9 81 80 21)(10 82 61 22)(11 83 62 23)(12 84 63 24)(13 85 64 25)(14 86 65 26)(15 87 66 27)(16 88 67 28)(17 89 68 29)(18 90 69 30)(19 91 70 31)(20 92 71 32)(41 137 115 156)(42 138 116 157)(43 139 117 158)(44 140 118 159)(45 121 119 160)(46 122 120 141)(47 123 101 142)(48 124 102 143)(49 125 103 144)(50 126 104 145)(51 127 105 146)(52 128 106 147)(53 129 107 148)(54 130 108 149)(55 131 109 150)(56 132 110 151)(57 133 111 152)(58 134 112 153)(59 135 113 154)(60 136 114 155)
(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 47 11 57)(2 56 12 46)(3 45 13 55)(4 54 14 44)(5 43 15 53)(6 52 16 42)(7 41 17 51)(8 50 18 60)(9 59 19 49)(10 48 20 58)(21 144 31 154)(22 153 32 143)(23 142 33 152)(24 151 34 141)(25 160 35 150)(26 149 36 159)(27 158 37 148)(28 147 38 157)(29 156 39 146)(30 145 40 155)(61 102 71 112)(62 111 72 101)(63 120 73 110)(64 109 74 119)(65 118 75 108)(66 107 76 117)(67 116 77 106)(68 105 78 115)(69 114 79 104)(70 103 80 113)(81 125 91 135)(82 134 92 124)(83 123 93 133)(84 132 94 122)(85 121 95 131)(86 130 96 140)(87 139 97 129)(88 128 98 138)(89 137 99 127)(90 126 100 136)
G:=sub<Sym(160)| (1,143,62,134)(2,135,63,144)(3,145,64,136)(4,137,65,146)(5,147,66,138)(6,139,67,148)(7,149,68,140)(8,121,69,150)(9,151,70,122)(10,123,71,152)(11,153,72,124)(12,125,73,154)(13,155,74,126)(14,127,75,156)(15,157,76,128)(16,129,77,158)(17,159,78,130)(18,131,79,160)(19,141,80,132)(20,133,61,142)(21,110,91,46)(22,47,92,111)(23,112,93,48)(24,49,94,113)(25,114,95,50)(26,51,96,115)(27,116,97,52)(28,53,98,117)(29,118,99,54)(30,55,100,119)(31,120,81,56)(32,57,82,101)(33,102,83,58)(34,59,84,103)(35,104,85,60)(36,41,86,105)(37,106,87,42)(38,43,88,107)(39,108,89,44)(40,45,90,109), (1,93,72,33)(2,94,73,34)(3,95,74,35)(4,96,75,36)(5,97,76,37)(6,98,77,38)(7,99,78,39)(8,100,79,40)(9,81,80,21)(10,82,61,22)(11,83,62,23)(12,84,63,24)(13,85,64,25)(14,86,65,26)(15,87,66,27)(16,88,67,28)(17,89,68,29)(18,90,69,30)(19,91,70,31)(20,92,71,32)(41,137,115,156)(42,138,116,157)(43,139,117,158)(44,140,118,159)(45,121,119,160)(46,122,120,141)(47,123,101,142)(48,124,102,143)(49,125,103,144)(50,126,104,145)(51,127,105,146)(52,128,106,147)(53,129,107,148)(54,130,108,149)(55,131,109,150)(56,132,110,151)(57,133,111,152)(58,134,112,153)(59,135,113,154)(60,136,114,155), (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), (1,47,11,57)(2,56,12,46)(3,45,13,55)(4,54,14,44)(5,43,15,53)(6,52,16,42)(7,41,17,51)(8,50,18,60)(9,59,19,49)(10,48,20,58)(21,144,31,154)(22,153,32,143)(23,142,33,152)(24,151,34,141)(25,160,35,150)(26,149,36,159)(27,158,37,148)(28,147,38,157)(29,156,39,146)(30,145,40,155)(61,102,71,112)(62,111,72,101)(63,120,73,110)(64,109,74,119)(65,118,75,108)(66,107,76,117)(67,116,77,106)(68,105,78,115)(69,114,79,104)(70,103,80,113)(81,125,91,135)(82,134,92,124)(83,123,93,133)(84,132,94,122)(85,121,95,131)(86,130,96,140)(87,139,97,129)(88,128,98,138)(89,137,99,127)(90,126,100,136)>;
G:=Group( (1,143,62,134)(2,135,63,144)(3,145,64,136)(4,137,65,146)(5,147,66,138)(6,139,67,148)(7,149,68,140)(8,121,69,150)(9,151,70,122)(10,123,71,152)(11,153,72,124)(12,125,73,154)(13,155,74,126)(14,127,75,156)(15,157,76,128)(16,129,77,158)(17,159,78,130)(18,131,79,160)(19,141,80,132)(20,133,61,142)(21,110,91,46)(22,47,92,111)(23,112,93,48)(24,49,94,113)(25,114,95,50)(26,51,96,115)(27,116,97,52)(28,53,98,117)(29,118,99,54)(30,55,100,119)(31,120,81,56)(32,57,82,101)(33,102,83,58)(34,59,84,103)(35,104,85,60)(36,41,86,105)(37,106,87,42)(38,43,88,107)(39,108,89,44)(40,45,90,109), (1,93,72,33)(2,94,73,34)(3,95,74,35)(4,96,75,36)(5,97,76,37)(6,98,77,38)(7,99,78,39)(8,100,79,40)(9,81,80,21)(10,82,61,22)(11,83,62,23)(12,84,63,24)(13,85,64,25)(14,86,65,26)(15,87,66,27)(16,88,67,28)(17,89,68,29)(18,90,69,30)(19,91,70,31)(20,92,71,32)(41,137,115,156)(42,138,116,157)(43,139,117,158)(44,140,118,159)(45,121,119,160)(46,122,120,141)(47,123,101,142)(48,124,102,143)(49,125,103,144)(50,126,104,145)(51,127,105,146)(52,128,106,147)(53,129,107,148)(54,130,108,149)(55,131,109,150)(56,132,110,151)(57,133,111,152)(58,134,112,153)(59,135,113,154)(60,136,114,155), (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), (1,47,11,57)(2,56,12,46)(3,45,13,55)(4,54,14,44)(5,43,15,53)(6,52,16,42)(7,41,17,51)(8,50,18,60)(9,59,19,49)(10,48,20,58)(21,144,31,154)(22,153,32,143)(23,142,33,152)(24,151,34,141)(25,160,35,150)(26,149,36,159)(27,158,37,148)(28,147,38,157)(29,156,39,146)(30,145,40,155)(61,102,71,112)(62,111,72,101)(63,120,73,110)(64,109,74,119)(65,118,75,108)(66,107,76,117)(67,116,77,106)(68,105,78,115)(69,114,79,104)(70,103,80,113)(81,125,91,135)(82,134,92,124)(83,123,93,133)(84,132,94,122)(85,121,95,131)(86,130,96,140)(87,139,97,129)(88,128,98,138)(89,137,99,127)(90,126,100,136) );
G=PermutationGroup([(1,143,62,134),(2,135,63,144),(3,145,64,136),(4,137,65,146),(5,147,66,138),(6,139,67,148),(7,149,68,140),(8,121,69,150),(9,151,70,122),(10,123,71,152),(11,153,72,124),(12,125,73,154),(13,155,74,126),(14,127,75,156),(15,157,76,128),(16,129,77,158),(17,159,78,130),(18,131,79,160),(19,141,80,132),(20,133,61,142),(21,110,91,46),(22,47,92,111),(23,112,93,48),(24,49,94,113),(25,114,95,50),(26,51,96,115),(27,116,97,52),(28,53,98,117),(29,118,99,54),(30,55,100,119),(31,120,81,56),(32,57,82,101),(33,102,83,58),(34,59,84,103),(35,104,85,60),(36,41,86,105),(37,106,87,42),(38,43,88,107),(39,108,89,44),(40,45,90,109)], [(1,93,72,33),(2,94,73,34),(3,95,74,35),(4,96,75,36),(5,97,76,37),(6,98,77,38),(7,99,78,39),(8,100,79,40),(9,81,80,21),(10,82,61,22),(11,83,62,23),(12,84,63,24),(13,85,64,25),(14,86,65,26),(15,87,66,27),(16,88,67,28),(17,89,68,29),(18,90,69,30),(19,91,70,31),(20,92,71,32),(41,137,115,156),(42,138,116,157),(43,139,117,158),(44,140,118,159),(45,121,119,160),(46,122,120,141),(47,123,101,142),(48,124,102,143),(49,125,103,144),(50,126,104,145),(51,127,105,146),(52,128,106,147),(53,129,107,148),(54,130,108,149),(55,131,109,150),(56,132,110,151),(57,133,111,152),(58,134,112,153),(59,135,113,154),(60,136,114,155)], [(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)], [(1,47,11,57),(2,56,12,46),(3,45,13,55),(4,54,14,44),(5,43,15,53),(6,52,16,42),(7,41,17,51),(8,50,18,60),(9,59,19,49),(10,48,20,58),(21,144,31,154),(22,153,32,143),(23,142,33,152),(24,151,34,141),(25,160,35,150),(26,149,36,159),(27,158,37,148),(28,147,38,157),(29,156,39,146),(30,145,40,155),(61,102,71,112),(62,111,72,101),(63,120,73,110),(64,109,74,119),(65,118,75,108),(66,107,76,117),(67,116,77,106),(68,105,78,115),(69,114,79,104),(70,103,80,113),(81,125,91,135),(82,134,92,124),(83,123,93,133),(84,132,94,122),(85,121,95,131),(86,130,96,140),(87,139,97,129),(88,128,98,138),(89,137,99,127),(90,126,100,136)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 28 | 28 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 20 | 21 | 0 | 0 |
| 18 | 21 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,9,0,0,0,0,9],[28,13,0,0,28,32,0,0,0,0,0,1,0,0,1,0],[20,18,0,0,21,21,0,0,0,0,0,40,0,0,40,0] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | Q8.10D10 | D5×C4○D4 |
| kernel | C42.132D10 | C20.6Q8 | C42⋊D5 | C4×D20 | C42⋊2D5 | Dic5.Q8 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊3Q8 | Q8×C20 | C4×Q8 | C20 | D10 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{132}D_{10} % in TeX
G:=Group("C4^2.132D10"); // GroupNames label
G:=SmallGroup(320,1253);
// by ID
G=gap.SmallGroup(320,1253);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀