metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.135D10, C10.672- (1+4), (C4×Q8)⋊17D5, (Q8×C20)⋊19C2, C4⋊C4.302D10, C20⋊2Q8⋊29C2, (C4×D20).23C2, D10⋊3Q8⋊11C2, (C4×Dic10)⋊41C2, C4.69(C4○D20), (C2×Q8).183D10, C20.122(C4○D4), (C4×C20).180C22, (C2×C10).128C24, (C2×C20).591C23, C4.50(Q8⋊2D5), C4.D20.13C2, (C2×D20).227C22, C4⋊Dic5.400C22, (Q8×C10).228C22, (C22×D5).50C23, C22.149(C23×D5), D10⋊C4.56C22, C5⋊4(C22.50C24), (C2×Dic5).228C23, (C4×Dic5).230C22, C2.25(D4.10D10), (C2×Dic10).251C22, C10.D4.157C22, C4⋊C4⋊7D5⋊18C2, C4⋊C4⋊D5⋊11C2, C2.67(C2×C4○D20), (C2×C4×D5).87C22, C10.113(C2×C4○D4), C2.13(C2×Q8⋊2D5), (C5×C4⋊C4).356C22, (C2×C4).172(C22×D5), SmallGroup(320,1256)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 694 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×6], C20 [×4], C20 [×5], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.50C24, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, Q8×C10, C4×Dic10 [×2], C20⋊2Q8, C4×D20, C4.D20 [×2], C4⋊C4⋊7D5 [×2], C4⋊C4⋊D5 [×4], D10⋊3Q8 [×2], Q8×C20, C42.135D10
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], C22.50C24, C4○D20 [×2], Q8⋊2D5 [×2], C23×D5, C2×C4○D20, C2×Q8⋊2D5, D4.10D10, C42.135D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c9 >
►On 160 points
Generators in S160
(1 118 43 125)(2 119 44 126)(3 120 45 127)(4 101 46 128)(5 102 47 129)(6 103 48 130)(7 104 49 131)(8 105 50 132)(9 106 51 133)(10 107 52 134)(11 108 53 135)(12 109 54 136)(13 110 55 137)(14 111 56 138)(15 112 57 139)(16 113 58 140)(17 114 59 121)(18 115 60 122)(19 116 41 123)(20 117 42 124)(21 158 75 99)(22 159 76 100)(23 160 77 81)(24 141 78 82)(25 142 79 83)(26 143 80 84)(27 144 61 85)(28 145 62 86)(29 146 63 87)(30 147 64 88)(31 148 65 89)(32 149 66 90)(33 150 67 91)(34 151 68 92)(35 152 69 93)(36 153 70 94)(37 154 71 95)(38 155 72 96)(39 156 73 97)(40 157 74 98)
(1 61 53 37)(2 72 54 28)(3 63 55 39)(4 74 56 30)(5 65 57 21)(6 76 58 32)(7 67 59 23)(8 78 60 34)(9 69 41 25)(10 80 42 36)(11 71 43 27)(12 62 44 38)(13 73 45 29)(14 64 46 40)(15 75 47 31)(16 66 48 22)(17 77 49 33)(18 68 50 24)(19 79 51 35)(20 70 52 26)(81 131 150 114)(82 122 151 105)(83 133 152 116)(84 124 153 107)(85 135 154 118)(86 126 155 109)(87 137 156 120)(88 128 157 111)(89 139 158 102)(90 130 159 113)(91 121 160 104)(92 132 141 115)(93 123 142 106)(94 134 143 117)(95 125 144 108)(96 136 145 119)(97 127 146 110)(98 138 147 101)(99 129 148 112)(100 140 149 103)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 32 31 22)(23 30 33 40)(24 39 34 29)(25 28 35 38)(26 37 36 27)(41 54 51 44)(42 43 52 53)(45 50 55 60)(46 59 56 49)(47 48 57 58)(61 80 71 70)(62 69 72 79)(63 78 73 68)(64 67 74 77)(65 76 75 66)(81 147 91 157)(82 156 92 146)(83 145 93 155)(84 154 94 144)(85 143 95 153)(86 152 96 142)(87 141 97 151)(88 150 98 160)(89 159 99 149)(90 148 100 158)(101 121 111 131)(102 130 112 140)(103 139 113 129)(104 128 114 138)(105 137 115 127)(106 126 116 136)(107 135 117 125)(108 124 118 134)(109 133 119 123)(110 122 120 132)
G:=sub<Sym(160)| (1,118,43,125)(2,119,44,126)(3,120,45,127)(4,101,46,128)(5,102,47,129)(6,103,48,130)(7,104,49,131)(8,105,50,132)(9,106,51,133)(10,107,52,134)(11,108,53,135)(12,109,54,136)(13,110,55,137)(14,111,56,138)(15,112,57,139)(16,113,58,140)(17,114,59,121)(18,115,60,122)(19,116,41,123)(20,117,42,124)(21,158,75,99)(22,159,76,100)(23,160,77,81)(24,141,78,82)(25,142,79,83)(26,143,80,84)(27,144,61,85)(28,145,62,86)(29,146,63,87)(30,147,64,88)(31,148,65,89)(32,149,66,90)(33,150,67,91)(34,151,68,92)(35,152,69,93)(36,153,70,94)(37,154,71,95)(38,155,72,96)(39,156,73,97)(40,157,74,98), (1,61,53,37)(2,72,54,28)(3,63,55,39)(4,74,56,30)(5,65,57,21)(6,76,58,32)(7,67,59,23)(8,78,60,34)(9,69,41,25)(10,80,42,36)(11,71,43,27)(12,62,44,38)(13,73,45,29)(14,64,46,40)(15,75,47,31)(16,66,48,22)(17,77,49,33)(18,68,50,24)(19,79,51,35)(20,70,52,26)(81,131,150,114)(82,122,151,105)(83,133,152,116)(84,124,153,107)(85,135,154,118)(86,126,155,109)(87,137,156,120)(88,128,157,111)(89,139,158,102)(90,130,159,113)(91,121,160,104)(92,132,141,115)(93,123,142,106)(94,134,143,117)(95,125,144,108)(96,136,145,119)(97,127,146,110)(98,138,147,101)(99,129,148,112)(100,140,149,103), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,32,31,22)(23,30,33,40)(24,39,34,29)(25,28,35,38)(26,37,36,27)(41,54,51,44)(42,43,52,53)(45,50,55,60)(46,59,56,49)(47,48,57,58)(61,80,71,70)(62,69,72,79)(63,78,73,68)(64,67,74,77)(65,76,75,66)(81,147,91,157)(82,156,92,146)(83,145,93,155)(84,154,94,144)(85,143,95,153)(86,152,96,142)(87,141,97,151)(88,150,98,160)(89,159,99,149)(90,148,100,158)(101,121,111,131)(102,130,112,140)(103,139,113,129)(104,128,114,138)(105,137,115,127)(106,126,116,136)(107,135,117,125)(108,124,118,134)(109,133,119,123)(110,122,120,132)>;
G:=Group( (1,118,43,125)(2,119,44,126)(3,120,45,127)(4,101,46,128)(5,102,47,129)(6,103,48,130)(7,104,49,131)(8,105,50,132)(9,106,51,133)(10,107,52,134)(11,108,53,135)(12,109,54,136)(13,110,55,137)(14,111,56,138)(15,112,57,139)(16,113,58,140)(17,114,59,121)(18,115,60,122)(19,116,41,123)(20,117,42,124)(21,158,75,99)(22,159,76,100)(23,160,77,81)(24,141,78,82)(25,142,79,83)(26,143,80,84)(27,144,61,85)(28,145,62,86)(29,146,63,87)(30,147,64,88)(31,148,65,89)(32,149,66,90)(33,150,67,91)(34,151,68,92)(35,152,69,93)(36,153,70,94)(37,154,71,95)(38,155,72,96)(39,156,73,97)(40,157,74,98), (1,61,53,37)(2,72,54,28)(3,63,55,39)(4,74,56,30)(5,65,57,21)(6,76,58,32)(7,67,59,23)(8,78,60,34)(9,69,41,25)(10,80,42,36)(11,71,43,27)(12,62,44,38)(13,73,45,29)(14,64,46,40)(15,75,47,31)(16,66,48,22)(17,77,49,33)(18,68,50,24)(19,79,51,35)(20,70,52,26)(81,131,150,114)(82,122,151,105)(83,133,152,116)(84,124,153,107)(85,135,154,118)(86,126,155,109)(87,137,156,120)(88,128,157,111)(89,139,158,102)(90,130,159,113)(91,121,160,104)(92,132,141,115)(93,123,142,106)(94,134,143,117)(95,125,144,108)(96,136,145,119)(97,127,146,110)(98,138,147,101)(99,129,148,112)(100,140,149,103), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,32,31,22)(23,30,33,40)(24,39,34,29)(25,28,35,38)(26,37,36,27)(41,54,51,44)(42,43,52,53)(45,50,55,60)(46,59,56,49)(47,48,57,58)(61,80,71,70)(62,69,72,79)(63,78,73,68)(64,67,74,77)(65,76,75,66)(81,147,91,157)(82,156,92,146)(83,145,93,155)(84,154,94,144)(85,143,95,153)(86,152,96,142)(87,141,97,151)(88,150,98,160)(89,159,99,149)(90,148,100,158)(101,121,111,131)(102,130,112,140)(103,139,113,129)(104,128,114,138)(105,137,115,127)(106,126,116,136)(107,135,117,125)(108,124,118,134)(109,133,119,123)(110,122,120,132) );
G=PermutationGroup([(1,118,43,125),(2,119,44,126),(3,120,45,127),(4,101,46,128),(5,102,47,129),(6,103,48,130),(7,104,49,131),(8,105,50,132),(9,106,51,133),(10,107,52,134),(11,108,53,135),(12,109,54,136),(13,110,55,137),(14,111,56,138),(15,112,57,139),(16,113,58,140),(17,114,59,121),(18,115,60,122),(19,116,41,123),(20,117,42,124),(21,158,75,99),(22,159,76,100),(23,160,77,81),(24,141,78,82),(25,142,79,83),(26,143,80,84),(27,144,61,85),(28,145,62,86),(29,146,63,87),(30,147,64,88),(31,148,65,89),(32,149,66,90),(33,150,67,91),(34,151,68,92),(35,152,69,93),(36,153,70,94),(37,154,71,95),(38,155,72,96),(39,156,73,97),(40,157,74,98)], [(1,61,53,37),(2,72,54,28),(3,63,55,39),(4,74,56,30),(5,65,57,21),(6,76,58,32),(7,67,59,23),(8,78,60,34),(9,69,41,25),(10,80,42,36),(11,71,43,27),(12,62,44,38),(13,73,45,29),(14,64,46,40),(15,75,47,31),(16,66,48,22),(17,77,49,33),(18,68,50,24),(19,79,51,35),(20,70,52,26),(81,131,150,114),(82,122,151,105),(83,133,152,116),(84,124,153,107),(85,135,154,118),(86,126,155,109),(87,137,156,120),(88,128,157,111),(89,139,158,102),(90,130,159,113),(91,121,160,104),(92,132,141,115),(93,123,142,106),(94,134,143,117),(95,125,144,108),(96,136,145,119),(97,127,146,110),(98,138,147,101),(99,129,148,112),(100,140,149,103)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,32,31,22),(23,30,33,40),(24,39,34,29),(25,28,35,38),(26,37,36,27),(41,54,51,44),(42,43,52,53),(45,50,55,60),(46,59,56,49),(47,48,57,58),(61,80,71,70),(62,69,72,79),(63,78,73,68),(64,67,74,77),(65,76,75,66),(81,147,91,157),(82,156,92,146),(83,145,93,155),(84,154,94,144),(85,143,95,153),(86,152,96,142),(87,141,97,151),(88,150,98,160),(89,159,99,149),(90,148,100,158),(101,121,111,131),(102,130,112,140),(103,139,113,129),(104,128,114,138),(105,137,115,127),(106,126,116,136),(107,135,117,125),(108,124,118,134),(109,133,119,123),(110,122,120,132)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 2 | 32 | 0 | 0 |
| 37 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 23 |
| 0 | 0 | 32 | 1 |
| 35 | 34 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 40 | 32 |
| 6 | 40 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [2,37,0,0,32,39,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,40,32,0,0,23,1],[35,6,0,0,34,0,0,0,0,0,9,40,0,0,0,32],[6,35,0,0,40,35,0,0,0,0,9,0,0,0,0,9] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | Q8⋊2D5 | D4.10D10 |
| kernel | C42.135D10 | C4×Dic10 | C20⋊2Q8 | C4×D20 | C4.D20 | C4⋊C4⋊7D5 | C4⋊C4⋊D5 | D10⋊3Q8 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 8 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{135}D_{10} % in TeX
G:=Group("C4^2.135D10"); // GroupNames label
G:=SmallGroup(320,1256);
// by ID
G=gap.SmallGroup(320,1256);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀