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), C42⋊C2.11D5, (C22×C4).362D10, C22.6(C2×Dic10), C22.94(C23×D5), C4⋊Dic5.360C22, C2.12(C22×Dic10), C23.150(C22×D5), C23.D5.91C22, (C22×C20).222C22, (C22×C10).131C23, Dic5.14D4.5C2, C5⋊2(C23.37C23), (C2×Dic5).203C23, (C4×Dic5).280C22, (C2×Dic10).236C22, C23.21D10.22C2, C10.D4.105C22, (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
Generators and relations for C42.88D10
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 >
Subgroups: 638 in 222 conjugacy classes, 115 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.37C23, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×C20, C4×Dic10, Dic5.14D4, C20⋊Q8, C4.Dic10, C2×C4×Dic5, C23.21D10, C5×C42⋊C2, C42.88D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, Dic10, C22×D5, C23.37C23, C2×Dic10, C23×D5, C22×Dic10, D5×C4○D4, C42.88D10
►On 160 points
Generators in S160
(1 95 62 130)(2 91 63 126)(3 97 64 122)(4 93 65 128)(5 99 61 124)(6 86 11 108)(7 82 12 104)(8 88 13 110)(9 84 14 106)(10 90 15 102)(16 103 77 81)(17 109 78 87)(18 105 79 83)(19 101 80 89)(20 107 76 85)(21 145 28 120)(22 141 29 116)(23 147 30 112)(24 143 26 118)(25 149 27 114)(31 94 39 129)(32 100 40 125)(33 96 36 121)(34 92 37 127)(35 98 38 123)(41 160 56 135)(42 156 57 131)(43 152 58 137)(44 158 59 133)(45 154 60 139)(46 155 51 140)(47 151 52 136)(48 157 53 132)(49 153 54 138)(50 159 55 134)(66 111 75 146)(67 117 71 142)(68 113 72 148)(69 119 73 144)(70 115 74 150)
(1 28 32 70)(2 29 33 66)(3 30 34 67)(4 26 35 68)(5 27 31 69)(6 41 77 46)(7 42 78 47)(8 43 79 48)(9 44 80 49)(10 45 76 50)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 40 74 62)(22 36 75 63)(23 37 71 64)(24 38 72 65)(25 39 73 61)(81 155 86 160)(82 156 87 151)(83 157 88 152)(84 158 89 153)(85 159 90 154)(91 116 96 111)(92 117 97 112)(93 118 98 113)(94 119 99 114)(95 120 100 115)(101 138 106 133)(102 139 107 134)(103 140 108 135)(104 131 109 136)(105 132 110 137)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)
(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 80 40 14)(2 79 36 13)(3 78 37 12)(4 77 38 11)(5 76 39 15)(6 65 16 35)(7 64 17 34)(8 63 18 33)(9 62 19 32)(10 61 20 31)(21 54 70 44)(22 53 66 43)(23 52 67 42)(24 51 68 41)(25 55 69 45)(26 46 72 56)(27 50 73 60)(28 49 74 59)(29 48 75 58)(30 47 71 57)(81 98 108 128)(82 97 109 127)(83 96 110 126)(84 95 101 125)(85 94 102 124)(86 93 103 123)(87 92 104 122)(88 91 105 121)(89 100 106 130)(90 99 107 129)(111 137 141 157)(112 136 142 156)(113 135 143 155)(114 134 144 154)(115 133 145 153)(116 132 146 152)(117 131 147 151)(118 140 148 160)(119 139 149 159)(120 138 150 158)
G:=sub<Sym(160)| (1,95,62,130)(2,91,63,126)(3,97,64,122)(4,93,65,128)(5,99,61,124)(6,86,11,108)(7,82,12,104)(8,88,13,110)(9,84,14,106)(10,90,15,102)(16,103,77,81)(17,109,78,87)(18,105,79,83)(19,101,80,89)(20,107,76,85)(21,145,28,120)(22,141,29,116)(23,147,30,112)(24,143,26,118)(25,149,27,114)(31,94,39,129)(32,100,40,125)(33,96,36,121)(34,92,37,127)(35,98,38,123)(41,160,56,135)(42,156,57,131)(43,152,58,137)(44,158,59,133)(45,154,60,139)(46,155,51,140)(47,151,52,136)(48,157,53,132)(49,153,54,138)(50,159,55,134)(66,111,75,146)(67,117,71,142)(68,113,72,148)(69,119,73,144)(70,115,74,150), (1,28,32,70)(2,29,33,66)(3,30,34,67)(4,26,35,68)(5,27,31,69)(6,41,77,46)(7,42,78,47)(8,43,79,48)(9,44,80,49)(10,45,76,50)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,40,74,62)(22,36,75,63)(23,37,71,64)(24,38,72,65)(25,39,73,61)(81,155,86,160)(82,156,87,151)(83,157,88,152)(84,158,89,153)(85,159,90,154)(91,116,96,111)(92,117,97,112)(93,118,98,113)(94,119,99,114)(95,120,100,115)(101,138,106,133)(102,139,107,134)(103,140,108,135)(104,131,109,136)(105,132,110,137)(121,146,126,141)(122,147,127,142)(123,148,128,143)(124,149,129,144)(125,150,130,145), (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), (1,80,40,14)(2,79,36,13)(3,78,37,12)(4,77,38,11)(5,76,39,15)(6,65,16,35)(7,64,17,34)(8,63,18,33)(9,62,19,32)(10,61,20,31)(21,54,70,44)(22,53,66,43)(23,52,67,42)(24,51,68,41)(25,55,69,45)(26,46,72,56)(27,50,73,60)(28,49,74,59)(29,48,75,58)(30,47,71,57)(81,98,108,128)(82,97,109,127)(83,96,110,126)(84,95,101,125)(85,94,102,124)(86,93,103,123)(87,92,104,122)(88,91,105,121)(89,100,106,130)(90,99,107,129)(111,137,141,157)(112,136,142,156)(113,135,143,155)(114,134,144,154)(115,133,145,153)(116,132,146,152)(117,131,147,151)(118,140,148,160)(119,139,149,159)(120,138,150,158)>;
G:=Group( (1,95,62,130)(2,91,63,126)(3,97,64,122)(4,93,65,128)(5,99,61,124)(6,86,11,108)(7,82,12,104)(8,88,13,110)(9,84,14,106)(10,90,15,102)(16,103,77,81)(17,109,78,87)(18,105,79,83)(19,101,80,89)(20,107,76,85)(21,145,28,120)(22,141,29,116)(23,147,30,112)(24,143,26,118)(25,149,27,114)(31,94,39,129)(32,100,40,125)(33,96,36,121)(34,92,37,127)(35,98,38,123)(41,160,56,135)(42,156,57,131)(43,152,58,137)(44,158,59,133)(45,154,60,139)(46,155,51,140)(47,151,52,136)(48,157,53,132)(49,153,54,138)(50,159,55,134)(66,111,75,146)(67,117,71,142)(68,113,72,148)(69,119,73,144)(70,115,74,150), (1,28,32,70)(2,29,33,66)(3,30,34,67)(4,26,35,68)(5,27,31,69)(6,41,77,46)(7,42,78,47)(8,43,79,48)(9,44,80,49)(10,45,76,50)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,40,74,62)(22,36,75,63)(23,37,71,64)(24,38,72,65)(25,39,73,61)(81,155,86,160)(82,156,87,151)(83,157,88,152)(84,158,89,153)(85,159,90,154)(91,116,96,111)(92,117,97,112)(93,118,98,113)(94,119,99,114)(95,120,100,115)(101,138,106,133)(102,139,107,134)(103,140,108,135)(104,131,109,136)(105,132,110,137)(121,146,126,141)(122,147,127,142)(123,148,128,143)(124,149,129,144)(125,150,130,145), (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), (1,80,40,14)(2,79,36,13)(3,78,37,12)(4,77,38,11)(5,76,39,15)(6,65,16,35)(7,64,17,34)(8,63,18,33)(9,62,19,32)(10,61,20,31)(21,54,70,44)(22,53,66,43)(23,52,67,42)(24,51,68,41)(25,55,69,45)(26,46,72,56)(27,50,73,60)(28,49,74,59)(29,48,75,58)(30,47,71,57)(81,98,108,128)(82,97,109,127)(83,96,110,126)(84,95,101,125)(85,94,102,124)(86,93,103,123)(87,92,104,122)(88,91,105,121)(89,100,106,130)(90,99,107,129)(111,137,141,157)(112,136,142,156)(113,135,143,155)(114,134,144,154)(115,133,145,153)(116,132,146,152)(117,131,147,151)(118,140,148,160)(119,139,149,159)(120,138,150,158) );
G=PermutationGroup([[(1,95,62,130),(2,91,63,126),(3,97,64,122),(4,93,65,128),(5,99,61,124),(6,86,11,108),(7,82,12,104),(8,88,13,110),(9,84,14,106),(10,90,15,102),(16,103,77,81),(17,109,78,87),(18,105,79,83),(19,101,80,89),(20,107,76,85),(21,145,28,120),(22,141,29,116),(23,147,30,112),(24,143,26,118),(25,149,27,114),(31,94,39,129),(32,100,40,125),(33,96,36,121),(34,92,37,127),(35,98,38,123),(41,160,56,135),(42,156,57,131),(43,152,58,137),(44,158,59,133),(45,154,60,139),(46,155,51,140),(47,151,52,136),(48,157,53,132),(49,153,54,138),(50,159,55,134),(66,111,75,146),(67,117,71,142),(68,113,72,148),(69,119,73,144),(70,115,74,150)], [(1,28,32,70),(2,29,33,66),(3,30,34,67),(4,26,35,68),(5,27,31,69),(6,41,77,46),(7,42,78,47),(8,43,79,48),(9,44,80,49),(10,45,76,50),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,40,74,62),(22,36,75,63),(23,37,71,64),(24,38,72,65),(25,39,73,61),(81,155,86,160),(82,156,87,151),(83,157,88,152),(84,158,89,153),(85,159,90,154),(91,116,96,111),(92,117,97,112),(93,118,98,113),(94,119,99,114),(95,120,100,115),(101,138,106,133),(102,139,107,134),(103,140,108,135),(104,131,109,136),(105,132,110,137),(121,146,126,141),(122,147,127,142),(123,148,128,143),(124,149,129,144),(125,150,130,145)], [(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)], [(1,80,40,14),(2,79,36,13),(3,78,37,12),(4,77,38,11),(5,76,39,15),(6,65,16,35),(7,64,17,34),(8,63,18,33),(9,62,19,32),(10,61,20,31),(21,54,70,44),(22,53,66,43),(23,52,67,42),(24,51,68,41),(25,55,69,45),(26,46,72,56),(27,50,73,60),(28,49,74,59),(29,48,75,58),(30,47,71,57),(81,98,108,128),(82,97,109,127),(83,96,110,126),(84,95,101,125),(85,94,102,124),(86,93,103,123),(87,92,104,122),(88,91,105,121),(89,100,106,130),(90,99,107,129),(111,137,141,157),(112,136,142,156),(113,135,143,155),(114,134,144,154),(115,133,145,153),(116,132,146,152),(117,131,147,151),(118,140,148,160),(119,139,149,159),(120,138,150,158)]])
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 |
Matrix representation of C42.88D10 ►in 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] >;
C42.88D10 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