metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.88D10, (C2×C20)⋊4Q8, C20⋊Q8⋊46C2, (C2×C4)⋊7Dic10, C20.68(C2×Q8), C4⋊C4.266D10, (C4×Dic10)⋊6C2, (C4×C20).21C22, (C2×C10).61C24, C22⋊C4.89D10, C4.Dic10⋊45C2, C4.33(C2×Dic10), C10.10(C22×Q8), (C2×C20).140C23, Dic5.7(C4○D4), (C22×C4).362D10, C42⋊C2.11D5, C22.6(C2×Dic10), C22.94(C23×D5), C4⋊Dic5.360C22, C2.12(C22×Dic10), C23.150(C22×D5), C23.D5.91C22, (C22×C10).131C23, (C22×C20).222C22, Dic5.14D4.5C2, C5⋊2(C23.37C23), (C4×Dic5).280C22, (C2×Dic5).203C23, (C2×Dic10).236C22, C10.D4.105C22, C23.21D10.22C2, (C22×Dic5).237C22, C2.8(D5×C4○D4), (C2×C10).12(C2×Q8), (C2×C4×Dic5).14C2, C10.130(C2×C4○D4), (C5×C4⋊C4).302C22, (C2×C4).574(C22×D5), (C5×C42⋊C2).12C2, (C5×C22⋊C4).98C22, SmallGroup(320,1189)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 638 in 222 conjugacy classes, 115 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×14], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C42⋊C2, C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.37C23, C4×Dic5 [×2], C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×6], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C4×Dic10 [×4], Dic5.14D4 [×4], C20⋊Q8 [×2], C4.Dic10 [×2], C2×C4×Dic5, C23.21D10, C5×C42⋊C2, C42.88D10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], Dic10 [×4], C22×D5 [×7], C23.37C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D5×C4○D4 [×2], C42.88D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 95 62 140)(2 91 63 136)(3 97 64 132)(4 93 65 138)(5 99 61 134)(6 86 11 118)(7 82 12 114)(8 88 13 120)(9 84 14 116)(10 90 15 112)(16 113 68 81)(17 119 69 87)(18 115 70 83)(19 111 66 89)(20 117 67 85)(21 143 26 158)(22 149 27 154)(23 145 28 160)(24 141 29 156)(25 147 30 152)(31 94 39 139)(32 100 40 135)(33 96 36 131)(34 92 37 137)(35 98 38 133)(41 101 56 130)(42 107 57 126)(43 103 58 122)(44 109 59 128)(45 105 60 124)(46 106 51 125)(47 102 52 121)(48 108 53 127)(49 104 54 123)(50 110 55 129)(71 144 79 159)(72 150 80 155)(73 146 76 151)(74 142 77 157)(75 148 78 153)
(1 72 32 23)(2 73 33 24)(3 74 34 25)(4 75 35 21)(5 71 31 22)(6 41 68 46)(7 42 69 47)(8 43 70 48)(9 44 66 49)(10 45 67 50)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(26 65 78 38)(27 61 79 39)(28 62 80 40)(29 63 76 36)(30 64 77 37)(81 106 86 101)(82 107 87 102)(83 108 88 103)(84 109 89 104)(85 110 90 105)(91 146 96 141)(92 147 97 142)(93 148 98 143)(94 149 99 144)(95 150 100 145)(111 123 116 128)(112 124 117 129)(113 125 118 130)(114 126 119 121)(115 127 120 122)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(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 66 40 14)(2 70 36 13)(3 69 37 12)(4 68 38 11)(5 67 39 15)(6 65 16 35)(7 64 17 34)(8 63 18 33)(9 62 19 32)(10 61 20 31)(21 41 78 51)(22 45 79 55)(23 44 80 54)(24 43 76 53)(25 42 77 52)(26 56 75 46)(27 60 71 50)(28 59 72 49)(29 58 73 48)(30 57 74 47)(81 98 118 138)(82 97 119 137)(83 96 120 136)(84 95 111 135)(85 94 112 134)(86 93 113 133)(87 92 114 132)(88 91 115 131)(89 100 116 140)(90 99 117 139)(101 148 125 158)(102 147 126 157)(103 146 127 156)(104 145 128 155)(105 144 129 154)(106 143 130 153)(107 142 121 152)(108 141 122 151)(109 150 123 160)(110 149 124 159)
G:=sub<Sym(160)| (1,95,62,140)(2,91,63,136)(3,97,64,132)(4,93,65,138)(5,99,61,134)(6,86,11,118)(7,82,12,114)(8,88,13,120)(9,84,14,116)(10,90,15,112)(16,113,68,81)(17,119,69,87)(18,115,70,83)(19,111,66,89)(20,117,67,85)(21,143,26,158)(22,149,27,154)(23,145,28,160)(24,141,29,156)(25,147,30,152)(31,94,39,139)(32,100,40,135)(33,96,36,131)(34,92,37,137)(35,98,38,133)(41,101,56,130)(42,107,57,126)(43,103,58,122)(44,109,59,128)(45,105,60,124)(46,106,51,125)(47,102,52,121)(48,108,53,127)(49,104,54,123)(50,110,55,129)(71,144,79,159)(72,150,80,155)(73,146,76,151)(74,142,77,157)(75,148,78,153), (1,72,32,23)(2,73,33,24)(3,74,34,25)(4,75,35,21)(5,71,31,22)(6,41,68,46)(7,42,69,47)(8,43,70,48)(9,44,66,49)(10,45,67,50)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(26,65,78,38)(27,61,79,39)(28,62,80,40)(29,63,76,36)(30,64,77,37)(81,106,86,101)(82,107,87,102)(83,108,88,103)(84,109,89,104)(85,110,90,105)(91,146,96,141)(92,147,97,142)(93,148,98,143)(94,149,99,144)(95,150,100,145)(111,123,116,128)(112,124,117,129)(113,125,118,130)(114,126,119,121)(115,127,120,122)(131,156,136,151)(132,157,137,152)(133,158,138,153)(134,159,139,154)(135,160,140,155), (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), (1,66,40,14)(2,70,36,13)(3,69,37,12)(4,68,38,11)(5,67,39,15)(6,65,16,35)(7,64,17,34)(8,63,18,33)(9,62,19,32)(10,61,20,31)(21,41,78,51)(22,45,79,55)(23,44,80,54)(24,43,76,53)(25,42,77,52)(26,56,75,46)(27,60,71,50)(28,59,72,49)(29,58,73,48)(30,57,74,47)(81,98,118,138)(82,97,119,137)(83,96,120,136)(84,95,111,135)(85,94,112,134)(86,93,113,133)(87,92,114,132)(88,91,115,131)(89,100,116,140)(90,99,117,139)(101,148,125,158)(102,147,126,157)(103,146,127,156)(104,145,128,155)(105,144,129,154)(106,143,130,153)(107,142,121,152)(108,141,122,151)(109,150,123,160)(110,149,124,159)>;
G:=Group( (1,95,62,140)(2,91,63,136)(3,97,64,132)(4,93,65,138)(5,99,61,134)(6,86,11,118)(7,82,12,114)(8,88,13,120)(9,84,14,116)(10,90,15,112)(16,113,68,81)(17,119,69,87)(18,115,70,83)(19,111,66,89)(20,117,67,85)(21,143,26,158)(22,149,27,154)(23,145,28,160)(24,141,29,156)(25,147,30,152)(31,94,39,139)(32,100,40,135)(33,96,36,131)(34,92,37,137)(35,98,38,133)(41,101,56,130)(42,107,57,126)(43,103,58,122)(44,109,59,128)(45,105,60,124)(46,106,51,125)(47,102,52,121)(48,108,53,127)(49,104,54,123)(50,110,55,129)(71,144,79,159)(72,150,80,155)(73,146,76,151)(74,142,77,157)(75,148,78,153), (1,72,32,23)(2,73,33,24)(3,74,34,25)(4,75,35,21)(5,71,31,22)(6,41,68,46)(7,42,69,47)(8,43,70,48)(9,44,66,49)(10,45,67,50)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(26,65,78,38)(27,61,79,39)(28,62,80,40)(29,63,76,36)(30,64,77,37)(81,106,86,101)(82,107,87,102)(83,108,88,103)(84,109,89,104)(85,110,90,105)(91,146,96,141)(92,147,97,142)(93,148,98,143)(94,149,99,144)(95,150,100,145)(111,123,116,128)(112,124,117,129)(113,125,118,130)(114,126,119,121)(115,127,120,122)(131,156,136,151)(132,157,137,152)(133,158,138,153)(134,159,139,154)(135,160,140,155), (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), (1,66,40,14)(2,70,36,13)(3,69,37,12)(4,68,38,11)(5,67,39,15)(6,65,16,35)(7,64,17,34)(8,63,18,33)(9,62,19,32)(10,61,20,31)(21,41,78,51)(22,45,79,55)(23,44,80,54)(24,43,76,53)(25,42,77,52)(26,56,75,46)(27,60,71,50)(28,59,72,49)(29,58,73,48)(30,57,74,47)(81,98,118,138)(82,97,119,137)(83,96,120,136)(84,95,111,135)(85,94,112,134)(86,93,113,133)(87,92,114,132)(88,91,115,131)(89,100,116,140)(90,99,117,139)(101,148,125,158)(102,147,126,157)(103,146,127,156)(104,145,128,155)(105,144,129,154)(106,143,130,153)(107,142,121,152)(108,141,122,151)(109,150,123,160)(110,149,124,159) );
G=PermutationGroup([(1,95,62,140),(2,91,63,136),(3,97,64,132),(4,93,65,138),(5,99,61,134),(6,86,11,118),(7,82,12,114),(8,88,13,120),(9,84,14,116),(10,90,15,112),(16,113,68,81),(17,119,69,87),(18,115,70,83),(19,111,66,89),(20,117,67,85),(21,143,26,158),(22,149,27,154),(23,145,28,160),(24,141,29,156),(25,147,30,152),(31,94,39,139),(32,100,40,135),(33,96,36,131),(34,92,37,137),(35,98,38,133),(41,101,56,130),(42,107,57,126),(43,103,58,122),(44,109,59,128),(45,105,60,124),(46,106,51,125),(47,102,52,121),(48,108,53,127),(49,104,54,123),(50,110,55,129),(71,144,79,159),(72,150,80,155),(73,146,76,151),(74,142,77,157),(75,148,78,153)], [(1,72,32,23),(2,73,33,24),(3,74,34,25),(4,75,35,21),(5,71,31,22),(6,41,68,46),(7,42,69,47),(8,43,70,48),(9,44,66,49),(10,45,67,50),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(26,65,78,38),(27,61,79,39),(28,62,80,40),(29,63,76,36),(30,64,77,37),(81,106,86,101),(82,107,87,102),(83,108,88,103),(84,109,89,104),(85,110,90,105),(91,146,96,141),(92,147,97,142),(93,148,98,143),(94,149,99,144),(95,150,100,145),(111,123,116,128),(112,124,117,129),(113,125,118,130),(114,126,119,121),(115,127,120,122),(131,156,136,151),(132,157,137,152),(133,158,138,153),(134,159,139,154),(135,160,140,155)], [(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)], [(1,66,40,14),(2,70,36,13),(3,69,37,12),(4,68,38,11),(5,67,39,15),(6,65,16,35),(7,64,17,34),(8,63,18,33),(9,62,19,32),(10,61,20,31),(21,41,78,51),(22,45,79,55),(23,44,80,54),(24,43,76,53),(25,42,77,52),(26,56,75,46),(27,60,71,50),(28,59,72,49),(29,58,73,48),(30,57,74,47),(81,98,118,138),(82,97,119,137),(83,96,120,136),(84,95,111,135),(85,94,112,134),(86,93,113,133),(87,92,114,132),(88,91,115,131),(89,100,116,140),(90,99,117,139),(101,148,125,158),(102,147,126,157),(103,146,127,156),(104,145,128,155),(105,144,129,154),(106,143,130,153),(107,142,121,152),(108,141,122,151),(109,150,123,160),(110,149,124,159)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 |
| 0 | 0 | 9 | 11 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 34 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 29 | 25 |
| 0 | 0 | 27 | 12 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,30,9,0,0,32,11],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,7,34,0,0,7,40],[1,0,0,0,0,40,0,0,0,0,29,27,0,0,25,12] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D10 | Dic10 | D5×C4○D4 |
| kernel | C42.88D10 | C4×Dic10 | Dic5.14D4 | C20⋊Q8 | C4.Dic10 | C2×C4×Dic5 | C23.21D10 | C5×C42⋊C2 | C2×C20 | C42⋊C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{88}D_{10} % in TeX
G:=Group("C4^2.88D10"); // GroupNames label
G:=SmallGroup(320,1189);
// by ID
G=gap.SmallGroup(320,1189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀