metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.276D10, (C2×C4)⋊8D20, (C4×D20)⋊3C2, (C2×C20)⋊31D4, C4⋊5(C4○D20), C4.90(C2×D20), (C2×C42)⋊11D5, C20⋊11(C4○D4), C4⋊D20⋊18C2, C20⋊7D4⋊50C2, C20⋊2Q8⋊38C2, C20.307(C2×D4), C22.6(C2×D20), C10.5(C22×D4), C2.7(C22×D20), C4.D20⋊33C2, (C2×C10).21C24, (C2×C20).694C23, (C4×C20).315C22, (C22×C4).440D10, (C2×Dic5).5C23, (C22×D5).3C23, C22.64(C23×D5), (C2×D20).211C22, C4⋊Dic5.289C22, C5⋊1(C22.26C24), C23.218(C22×D5), D10⋊C4.80C22, (C22×C20).524C22, (C22×C10).383C23, (C2×Dic10).232C22, (C2×C4×C20)⋊13C2, (C2×C4○D20)⋊2C2, C10.8(C2×C4○D4), C2.10(C2×C4○D20), (C2×C10).172(C2×D4), (C2×C4×D5).238C22, (C2×C4).730(C22×D5), (C2×C5⋊D4).93C22, SmallGroup(320,1149)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1182 in 310 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×20], Q8 [×4], C23, C23 [×4], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], C20 [×8], C20 [×2], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×8], D20 [×12], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×D5 [×4], C22×C10, C22.26C24, C4⋊Dic5 [×4], D10⋊C4 [×8], C4×C20 [×2], C4×C20 [×2], C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×6], C4○D20 [×8], C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], C20⋊2Q8, C4×D20 [×4], C4⋊D20, C4.D20 [×2], C20⋊7D4 [×4], C2×C4×C20, C2×C4○D20 [×2], C42.276D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], D20 [×4], C22×D5 [×7], C22.26C24, C2×D20 [×6], C4○D20 [×4], C23×D5, C22×D20, C2×C4○D20 [×2], C42.276D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >
►On 160 points
Generators in S160
(1 113 33 92)(2 114 34 93)(3 115 35 94)(4 116 36 95)(5 117 37 96)(6 118 38 97)(7 119 39 98)(8 120 40 99)(9 111 31 100)(10 112 32 91)(11 129 49 109)(12 130 50 110)(13 121 41 101)(14 122 42 102)(15 123 43 103)(16 124 44 104)(17 125 45 105)(18 126 46 106)(19 127 47 107)(20 128 48 108)(21 85 145 65)(22 86 146 66)(23 87 147 67)(24 88 148 68)(25 89 149 69)(26 90 150 70)(27 81 141 61)(28 82 142 62)(29 83 143 63)(30 84 144 64)(51 152 72 131)(52 153 73 132)(53 154 74 133)(54 155 75 134)(55 156 76 135)(56 157 77 136)(57 158 78 137)(58 159 79 138)(59 160 80 139)(60 151 71 140)
(1 70 12 52)(2 61 13 53)(3 62 14 54)(4 63 15 55)(5 64 16 56)(6 65 17 57)(7 66 18 58)(8 67 19 59)(9 68 20 60)(10 69 11 51)(21 125 158 118)(22 126 159 119)(23 127 160 120)(24 128 151 111)(25 129 152 112)(26 130 153 113)(27 121 154 114)(28 122 155 115)(29 123 156 116)(30 124 157 117)(31 88 48 71)(32 89 49 72)(33 90 50 73)(34 81 41 74)(35 82 42 75)(36 83 43 76)(37 84 44 77)(38 85 45 78)(39 86 46 79)(40 87 47 80)(91 149 109 131)(92 150 110 132)(93 141 101 133)(94 142 102 134)(95 143 103 135)(96 144 104 136)(97 145 105 137)(98 146 106 138)(99 147 107 139)(100 148 108 140)
(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 91 12 109)(2 108 13 100)(3 99 14 107)(4 106 15 98)(5 97 16 105)(6 104 17 96)(7 95 18 103)(8 102 19 94)(9 93 20 101)(10 110 11 92)(21 77 158 84)(22 83 159 76)(23 75 160 82)(24 81 151 74)(25 73 152 90)(26 89 153 72)(27 71 154 88)(28 87 155 80)(29 79 156 86)(30 85 157 78)(31 114 48 121)(32 130 49 113)(33 112 50 129)(34 128 41 111)(35 120 42 127)(36 126 43 119)(37 118 44 125)(38 124 45 117)(39 116 46 123)(40 122 47 115)(51 150 69 132)(52 131 70 149)(53 148 61 140)(54 139 62 147)(55 146 63 138)(56 137 64 145)(57 144 65 136)(58 135 66 143)(59 142 67 134)(60 133 68 141)
G:=sub<Sym(160)| (1,113,33,92)(2,114,34,93)(3,115,35,94)(4,116,36,95)(5,117,37,96)(6,118,38,97)(7,119,39,98)(8,120,40,99)(9,111,31,100)(10,112,32,91)(11,129,49,109)(12,130,50,110)(13,121,41,101)(14,122,42,102)(15,123,43,103)(16,124,44,104)(17,125,45,105)(18,126,46,106)(19,127,47,107)(20,128,48,108)(21,85,145,65)(22,86,146,66)(23,87,147,67)(24,88,148,68)(25,89,149,69)(26,90,150,70)(27,81,141,61)(28,82,142,62)(29,83,143,63)(30,84,144,64)(51,152,72,131)(52,153,73,132)(53,154,74,133)(54,155,75,134)(55,156,76,135)(56,157,77,136)(57,158,78,137)(58,159,79,138)(59,160,80,139)(60,151,71,140), (1,70,12,52)(2,61,13,53)(3,62,14,54)(4,63,15,55)(5,64,16,56)(6,65,17,57)(7,66,18,58)(8,67,19,59)(9,68,20,60)(10,69,11,51)(21,125,158,118)(22,126,159,119)(23,127,160,120)(24,128,151,111)(25,129,152,112)(26,130,153,113)(27,121,154,114)(28,122,155,115)(29,123,156,116)(30,124,157,117)(31,88,48,71)(32,89,49,72)(33,90,50,73)(34,81,41,74)(35,82,42,75)(36,83,43,76)(37,84,44,77)(38,85,45,78)(39,86,46,79)(40,87,47,80)(91,149,109,131)(92,150,110,132)(93,141,101,133)(94,142,102,134)(95,143,103,135)(96,144,104,136)(97,145,105,137)(98,146,106,138)(99,147,107,139)(100,148,108,140), (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), (1,91,12,109)(2,108,13,100)(3,99,14,107)(4,106,15,98)(5,97,16,105)(6,104,17,96)(7,95,18,103)(8,102,19,94)(9,93,20,101)(10,110,11,92)(21,77,158,84)(22,83,159,76)(23,75,160,82)(24,81,151,74)(25,73,152,90)(26,89,153,72)(27,71,154,88)(28,87,155,80)(29,79,156,86)(30,85,157,78)(31,114,48,121)(32,130,49,113)(33,112,50,129)(34,128,41,111)(35,120,42,127)(36,126,43,119)(37,118,44,125)(38,124,45,117)(39,116,46,123)(40,122,47,115)(51,150,69,132)(52,131,70,149)(53,148,61,140)(54,139,62,147)(55,146,63,138)(56,137,64,145)(57,144,65,136)(58,135,66,143)(59,142,67,134)(60,133,68,141)>;
G:=Group( (1,113,33,92)(2,114,34,93)(3,115,35,94)(4,116,36,95)(5,117,37,96)(6,118,38,97)(7,119,39,98)(8,120,40,99)(9,111,31,100)(10,112,32,91)(11,129,49,109)(12,130,50,110)(13,121,41,101)(14,122,42,102)(15,123,43,103)(16,124,44,104)(17,125,45,105)(18,126,46,106)(19,127,47,107)(20,128,48,108)(21,85,145,65)(22,86,146,66)(23,87,147,67)(24,88,148,68)(25,89,149,69)(26,90,150,70)(27,81,141,61)(28,82,142,62)(29,83,143,63)(30,84,144,64)(51,152,72,131)(52,153,73,132)(53,154,74,133)(54,155,75,134)(55,156,76,135)(56,157,77,136)(57,158,78,137)(58,159,79,138)(59,160,80,139)(60,151,71,140), (1,70,12,52)(2,61,13,53)(3,62,14,54)(4,63,15,55)(5,64,16,56)(6,65,17,57)(7,66,18,58)(8,67,19,59)(9,68,20,60)(10,69,11,51)(21,125,158,118)(22,126,159,119)(23,127,160,120)(24,128,151,111)(25,129,152,112)(26,130,153,113)(27,121,154,114)(28,122,155,115)(29,123,156,116)(30,124,157,117)(31,88,48,71)(32,89,49,72)(33,90,50,73)(34,81,41,74)(35,82,42,75)(36,83,43,76)(37,84,44,77)(38,85,45,78)(39,86,46,79)(40,87,47,80)(91,149,109,131)(92,150,110,132)(93,141,101,133)(94,142,102,134)(95,143,103,135)(96,144,104,136)(97,145,105,137)(98,146,106,138)(99,147,107,139)(100,148,108,140), (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), (1,91,12,109)(2,108,13,100)(3,99,14,107)(4,106,15,98)(5,97,16,105)(6,104,17,96)(7,95,18,103)(8,102,19,94)(9,93,20,101)(10,110,11,92)(21,77,158,84)(22,83,159,76)(23,75,160,82)(24,81,151,74)(25,73,152,90)(26,89,153,72)(27,71,154,88)(28,87,155,80)(29,79,156,86)(30,85,157,78)(31,114,48,121)(32,130,49,113)(33,112,50,129)(34,128,41,111)(35,120,42,127)(36,126,43,119)(37,118,44,125)(38,124,45,117)(39,116,46,123)(40,122,47,115)(51,150,69,132)(52,131,70,149)(53,148,61,140)(54,139,62,147)(55,146,63,138)(56,137,64,145)(57,144,65,136)(58,135,66,143)(59,142,67,134)(60,133,68,141) );
G=PermutationGroup([(1,113,33,92),(2,114,34,93),(3,115,35,94),(4,116,36,95),(5,117,37,96),(6,118,38,97),(7,119,39,98),(8,120,40,99),(9,111,31,100),(10,112,32,91),(11,129,49,109),(12,130,50,110),(13,121,41,101),(14,122,42,102),(15,123,43,103),(16,124,44,104),(17,125,45,105),(18,126,46,106),(19,127,47,107),(20,128,48,108),(21,85,145,65),(22,86,146,66),(23,87,147,67),(24,88,148,68),(25,89,149,69),(26,90,150,70),(27,81,141,61),(28,82,142,62),(29,83,143,63),(30,84,144,64),(51,152,72,131),(52,153,73,132),(53,154,74,133),(54,155,75,134),(55,156,76,135),(56,157,77,136),(57,158,78,137),(58,159,79,138),(59,160,80,139),(60,151,71,140)], [(1,70,12,52),(2,61,13,53),(3,62,14,54),(4,63,15,55),(5,64,16,56),(6,65,17,57),(7,66,18,58),(8,67,19,59),(9,68,20,60),(10,69,11,51),(21,125,158,118),(22,126,159,119),(23,127,160,120),(24,128,151,111),(25,129,152,112),(26,130,153,113),(27,121,154,114),(28,122,155,115),(29,123,156,116),(30,124,157,117),(31,88,48,71),(32,89,49,72),(33,90,50,73),(34,81,41,74),(35,82,42,75),(36,83,43,76),(37,84,44,77),(38,85,45,78),(39,86,46,79),(40,87,47,80),(91,149,109,131),(92,150,110,132),(93,141,101,133),(94,142,102,134),(95,143,103,135),(96,144,104,136),(97,145,105,137),(98,146,106,138),(99,147,107,139),(100,148,108,140)], [(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)], [(1,91,12,109),(2,108,13,100),(3,99,14,107),(4,106,15,98),(5,97,16,105),(6,104,17,96),(7,95,18,103),(8,102,19,94),(9,93,20,101),(10,110,11,92),(21,77,158,84),(22,83,159,76),(23,75,160,82),(24,81,151,74),(25,73,152,90),(26,89,153,72),(27,71,154,88),(28,87,155,80),(29,79,156,86),(30,85,157,78),(31,114,48,121),(32,130,49,113),(33,112,50,129),(34,128,41,111),(35,120,42,127),(36,126,43,119),(37,118,44,125),(38,124,45,117),(39,116,46,123),(40,122,47,115),(51,150,69,132),(52,131,70,149),(53,148,61,140),(54,139,62,147),(55,146,63,138),(56,137,64,145),(57,144,65,136),(58,135,66,143),(59,142,67,134),(60,133,68,141)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 30 | 28 |
| 0 | 0 | 22 | 11 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 37 | 0 | 0 |
| 21 | 38 | 0 | 0 |
| 0 | 0 | 3 | 21 |
| 0 | 0 | 37 | 40 |
| 38 | 4 | 0 | 0 |
| 18 | 3 | 0 | 0 |
| 0 | 0 | 19 | 32 |
| 0 | 0 | 22 | 22 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,30,22,0,0,28,11],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,21,0,0,37,38,0,0,0,0,3,37,0,0,21,40],[38,18,0,0,4,3,0,0,0,0,19,22,0,0,32,22] >;
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C4○D20 |
| kernel | C42.276D10 | C20⋊2Q8 | C4×D20 | C4⋊D20 | C4.D20 | C20⋊7D4 | C2×C4×C20 | C2×C4○D20 | C2×C20 | C2×C42 | C20 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 2 | 4 | 1 | 2 | 4 | 2 | 8 | 8 | 6 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_4^2._{276}D_{10} % in TeX
G:=Group("C4^2.276D10"); // GroupNames label
G:=SmallGroup(320,1149);
// by ID
G=gap.SmallGroup(320,1149);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,12550]);
// 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^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀