metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.151D10, C10.292- (1+4), C42.C2⋊7D5, C4⋊C4.112D10, D10⋊Q8⋊36C2, C42⋊D5⋊37C2, (C2×C20).89C23, Dic5⋊3Q8⋊36C2, D10.38(C4○D4), D20⋊8C4.12C2, (C4×C20).240C22, (C2×C10).237C24, Dic5.46(C4○D4), Dic5.Q8⋊34C2, D10.13D4.2C2, (C2×D20).171C22, C4⋊Dic5.242C22, C22.258(C23×D5), C5⋊9(C22.46C24), (C2×Dic5).269C23, (C4×Dic5).235C22, C10.D4.53C22, (C22×D5).232C23, D10⋊C4.137C22, C2.30(Q8.10D10), (C2×Dic10).187C22, (D5×C4⋊C4)⋊37C2, C2.88(D5×C4○D4), C4⋊C4⋊D5⋊35C2, C4⋊C4⋊7D5⋊36C2, C10.199(C2×C4○D4), (C5×C42.C2)⋊10C2, (C2×C4×D5).136C22, (C5×C4⋊C4).192C22, (C2×C4).204(C22×D5), SmallGroup(320,1365)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 710 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×7], C5, C2×C4 [×7], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×7], D10 [×2], D10 [×5], C2×C10, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C2×C20 [×7], C22×D5 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×6], C2×Dic10, C2×C4×D5 [×4], C2×D20, C42⋊D5 [×2], Dic5⋊3Q8, Dic5.Q8 [×2], D5×C4⋊C4, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.151D10
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, C23×D5, Q8.10D10, D5×C4○D4 [×2], C42.151D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c9 >
(1 148 11 158)(2 126 12 136)(3 150 13 160)(4 128 14 138)(5 152 15 142)(6 130 16 140)(7 154 17 144)(8 132 18 122)(9 156 19 146)(10 134 20 124)(21 116 31 106)(22 56 32 46)(23 118 33 108)(24 58 34 48)(25 120 35 110)(26 60 36 50)(27 102 37 112)(28 42 38 52)(29 104 39 114)(30 44 40 54)(41 83 51 93)(43 85 53 95)(45 87 55 97)(47 89 57 99)(49 91 59 81)(61 149 71 159)(62 127 72 137)(63 151 73 141)(64 129 74 139)(65 153 75 143)(66 131 76 121)(67 155 77 145)(68 133 78 123)(69 157 79 147)(70 135 80 125)(82 111 92 101)(84 113 94 103)(86 115 96 105)(88 117 98 107)(90 119 100 109)
(1 52 80 113)(2 43 61 104)(3 54 62 115)(4 45 63 106)(5 56 64 117)(6 47 65 108)(7 58 66 119)(8 49 67 110)(9 60 68 101)(10 51 69 112)(11 42 70 103)(12 53 71 114)(13 44 72 105)(14 55 73 116)(15 46 74 107)(16 57 75 118)(17 48 76 109)(18 59 77 120)(19 50 78 111)(20 41 79 102)(21 128 87 151)(22 139 88 142)(23 130 89 153)(24 121 90 144)(25 132 91 155)(26 123 92 146)(27 134 93 157)(28 125 94 148)(29 136 95 159)(30 127 96 150)(31 138 97 141)(32 129 98 152)(33 140 99 143)(34 131 100 154)(35 122 81 145)(36 133 82 156)(37 124 83 147)(38 135 84 158)(39 126 85 149)(40 137 86 160)
(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)
(1 32 11 22)(2 21 12 31)(3 30 13 40)(4 39 14 29)(5 28 15 38)(6 37 16 27)(7 26 17 36)(8 35 18 25)(9 24 19 34)(10 33 20 23)(41 130 51 140)(42 139 52 129)(43 128 53 138)(44 137 54 127)(45 126 55 136)(46 135 56 125)(47 124 57 134)(48 133 58 123)(49 122 59 132)(50 131 60 121)(61 87 71 97)(62 96 72 86)(63 85 73 95)(64 94 74 84)(65 83 75 93)(66 92 76 82)(67 81 77 91)(68 90 78 100)(69 99 79 89)(70 88 80 98)(101 144 111 154)(102 153 112 143)(103 142 113 152)(104 151 114 141)(105 160 115 150)(106 149 116 159)(107 158 117 148)(108 147 118 157)(109 156 119 146)(110 145 120 155)
G:=sub<Sym(160)| (1,148,11,158)(2,126,12,136)(3,150,13,160)(4,128,14,138)(5,152,15,142)(6,130,16,140)(7,154,17,144)(8,132,18,122)(9,156,19,146)(10,134,20,124)(21,116,31,106)(22,56,32,46)(23,118,33,108)(24,58,34,48)(25,120,35,110)(26,60,36,50)(27,102,37,112)(28,42,38,52)(29,104,39,114)(30,44,40,54)(41,83,51,93)(43,85,53,95)(45,87,55,97)(47,89,57,99)(49,91,59,81)(61,149,71,159)(62,127,72,137)(63,151,73,141)(64,129,74,139)(65,153,75,143)(66,131,76,121)(67,155,77,145)(68,133,78,123)(69,157,79,147)(70,135,80,125)(82,111,92,101)(84,113,94,103)(86,115,96,105)(88,117,98,107)(90,119,100,109), (1,52,80,113)(2,43,61,104)(3,54,62,115)(4,45,63,106)(5,56,64,117)(6,47,65,108)(7,58,66,119)(8,49,67,110)(9,60,68,101)(10,51,69,112)(11,42,70,103)(12,53,71,114)(13,44,72,105)(14,55,73,116)(15,46,74,107)(16,57,75,118)(17,48,76,109)(18,59,77,120)(19,50,78,111)(20,41,79,102)(21,128,87,151)(22,139,88,142)(23,130,89,153)(24,121,90,144)(25,132,91,155)(26,123,92,146)(27,134,93,157)(28,125,94,148)(29,136,95,159)(30,127,96,150)(31,138,97,141)(32,129,98,152)(33,140,99,143)(34,131,100,154)(35,122,81,145)(36,133,82,156)(37,124,83,147)(38,135,84,158)(39,126,85,149)(40,137,86,160), (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), (1,32,11,22)(2,21,12,31)(3,30,13,40)(4,39,14,29)(5,28,15,38)(6,37,16,27)(7,26,17,36)(8,35,18,25)(9,24,19,34)(10,33,20,23)(41,130,51,140)(42,139,52,129)(43,128,53,138)(44,137,54,127)(45,126,55,136)(46,135,56,125)(47,124,57,134)(48,133,58,123)(49,122,59,132)(50,131,60,121)(61,87,71,97)(62,96,72,86)(63,85,73,95)(64,94,74,84)(65,83,75,93)(66,92,76,82)(67,81,77,91)(68,90,78,100)(69,99,79,89)(70,88,80,98)(101,144,111,154)(102,153,112,143)(103,142,113,152)(104,151,114,141)(105,160,115,150)(106,149,116,159)(107,158,117,148)(108,147,118,157)(109,156,119,146)(110,145,120,155)>;
G:=Group( (1,148,11,158)(2,126,12,136)(3,150,13,160)(4,128,14,138)(5,152,15,142)(6,130,16,140)(7,154,17,144)(8,132,18,122)(9,156,19,146)(10,134,20,124)(21,116,31,106)(22,56,32,46)(23,118,33,108)(24,58,34,48)(25,120,35,110)(26,60,36,50)(27,102,37,112)(28,42,38,52)(29,104,39,114)(30,44,40,54)(41,83,51,93)(43,85,53,95)(45,87,55,97)(47,89,57,99)(49,91,59,81)(61,149,71,159)(62,127,72,137)(63,151,73,141)(64,129,74,139)(65,153,75,143)(66,131,76,121)(67,155,77,145)(68,133,78,123)(69,157,79,147)(70,135,80,125)(82,111,92,101)(84,113,94,103)(86,115,96,105)(88,117,98,107)(90,119,100,109), (1,52,80,113)(2,43,61,104)(3,54,62,115)(4,45,63,106)(5,56,64,117)(6,47,65,108)(7,58,66,119)(8,49,67,110)(9,60,68,101)(10,51,69,112)(11,42,70,103)(12,53,71,114)(13,44,72,105)(14,55,73,116)(15,46,74,107)(16,57,75,118)(17,48,76,109)(18,59,77,120)(19,50,78,111)(20,41,79,102)(21,128,87,151)(22,139,88,142)(23,130,89,153)(24,121,90,144)(25,132,91,155)(26,123,92,146)(27,134,93,157)(28,125,94,148)(29,136,95,159)(30,127,96,150)(31,138,97,141)(32,129,98,152)(33,140,99,143)(34,131,100,154)(35,122,81,145)(36,133,82,156)(37,124,83,147)(38,135,84,158)(39,126,85,149)(40,137,86,160), (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), (1,32,11,22)(2,21,12,31)(3,30,13,40)(4,39,14,29)(5,28,15,38)(6,37,16,27)(7,26,17,36)(8,35,18,25)(9,24,19,34)(10,33,20,23)(41,130,51,140)(42,139,52,129)(43,128,53,138)(44,137,54,127)(45,126,55,136)(46,135,56,125)(47,124,57,134)(48,133,58,123)(49,122,59,132)(50,131,60,121)(61,87,71,97)(62,96,72,86)(63,85,73,95)(64,94,74,84)(65,83,75,93)(66,92,76,82)(67,81,77,91)(68,90,78,100)(69,99,79,89)(70,88,80,98)(101,144,111,154)(102,153,112,143)(103,142,113,152)(104,151,114,141)(105,160,115,150)(106,149,116,159)(107,158,117,148)(108,147,118,157)(109,156,119,146)(110,145,120,155) );
G=PermutationGroup([(1,148,11,158),(2,126,12,136),(3,150,13,160),(4,128,14,138),(5,152,15,142),(6,130,16,140),(7,154,17,144),(8,132,18,122),(9,156,19,146),(10,134,20,124),(21,116,31,106),(22,56,32,46),(23,118,33,108),(24,58,34,48),(25,120,35,110),(26,60,36,50),(27,102,37,112),(28,42,38,52),(29,104,39,114),(30,44,40,54),(41,83,51,93),(43,85,53,95),(45,87,55,97),(47,89,57,99),(49,91,59,81),(61,149,71,159),(62,127,72,137),(63,151,73,141),(64,129,74,139),(65,153,75,143),(66,131,76,121),(67,155,77,145),(68,133,78,123),(69,157,79,147),(70,135,80,125),(82,111,92,101),(84,113,94,103),(86,115,96,105),(88,117,98,107),(90,119,100,109)], [(1,52,80,113),(2,43,61,104),(3,54,62,115),(4,45,63,106),(5,56,64,117),(6,47,65,108),(7,58,66,119),(8,49,67,110),(9,60,68,101),(10,51,69,112),(11,42,70,103),(12,53,71,114),(13,44,72,105),(14,55,73,116),(15,46,74,107),(16,57,75,118),(17,48,76,109),(18,59,77,120),(19,50,78,111),(20,41,79,102),(21,128,87,151),(22,139,88,142),(23,130,89,153),(24,121,90,144),(25,132,91,155),(26,123,92,146),(27,134,93,157),(28,125,94,148),(29,136,95,159),(30,127,96,150),(31,138,97,141),(32,129,98,152),(33,140,99,143),(34,131,100,154),(35,122,81,145),(36,133,82,156),(37,124,83,147),(38,135,84,158),(39,126,85,149),(40,137,86,160)], [(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)], [(1,32,11,22),(2,21,12,31),(3,30,13,40),(4,39,14,29),(5,28,15,38),(6,37,16,27),(7,26,17,36),(8,35,18,25),(9,24,19,34),(10,33,20,23),(41,130,51,140),(42,139,52,129),(43,128,53,138),(44,137,54,127),(45,126,55,136),(46,135,56,125),(47,124,57,134),(48,133,58,123),(49,122,59,132),(50,131,60,121),(61,87,71,97),(62,96,72,86),(63,85,73,95),(64,94,74,84),(65,83,75,93),(66,92,76,82),(67,81,77,91),(68,90,78,100),(69,99,79,89),(70,88,80,98),(101,144,111,154),(102,153,112,143),(103,142,113,152),(104,151,114,141),(105,160,115,150),(106,149,116,159),(107,158,117,148),(108,147,118,157),(109,156,119,146),(110,145,120,155)])
Matrix representation ►G ⊆ GL6(𝔽41)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 32 |
0 | 0 | 0 | 0 | 9 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 9 | 0 |
0 | 0 | 0 | 0 | 0 | 9 |
1 | 34 | 0 | 0 | 0 | 0 |
7 | 34 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
40 | 0 | 0 | 0 | 0 | 0 |
34 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 32 | 0 | 0 | 0 |
0 | 0 | 0 | 32 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,7,0,0,0,0,34,34,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
53 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 2- (1+4) | Q8.10D10 | D5×C4○D4 |
kernel | C42.151D10 | C42⋊D5 | Dic5⋊3Q8 | Dic5.Q8 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | Dic5 | D10 | C42 | C4⋊C4 | C10 | C2 | C2 |
# reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 12 | 1 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{151}D_{10}
% in TeX
G:=Group("C4^2.151D10");
// GroupNames label
G:=SmallGroup(320,1365);
// by ID
G=gap.SmallGroup(320,1365);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,346,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations