metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.236D10, (C2×C20).450D4, C42⋊C2⋊3D5, D20⋊6C4⋊28C2, C10.86(C4○D8), C4.91(C4○D20), C20⋊7D4.11C2, C20.55D4⋊6C2, C10.D8⋊29C2, C20.Q8⋊29C2, (C22×C10).76D4, C20.179(C4○D4), C2.6(D4⋊D10), (C2×C20).330C23, (C2×D20).96C22, (C22×C4).111D10, C5⋊6(C23.19D4), C23.20(C5⋊D4), C10.106(C8⋊C22), C2.9(D4.8D10), C4⋊Dic5.135C22, (C22×C20).152C22, C10.66(C22.D4), C2.17(C23.23D10), (C5×C42⋊C2)⋊3C2, (C2×C10).459(C2×D4), (C2×C4).215(C5⋊D4), (C5×C4⋊C4).267C22, (C2×C5⋊2C8).87C22, (C2×C4).430(C22×D5), C22.145(C2×C5⋊D4), SmallGroup(320,630)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.236D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c9 >
Subgroups: 446 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.19D4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C10.D8, C20.Q8, D20⋊6C4, C20.55D4, C20⋊7D4, C5×C42⋊C2, C4⋊C4.236D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, C5⋊D4, C22×D5, C23.19D4, C4○D20, C2×C5⋊D4, C23.23D10, D4⋊D10, D4.8D10, C4⋊C4.236D10
►On 160 points
Generators in S160
(1 105 11 115)(2 106 12 116)(3 107 13 117)(4 108 14 118)(5 109 15 119)(6 110 16 120)(7 111 17 101)(8 112 18 102)(9 113 19 103)(10 114 20 104)(21 150 31 160)(22 151 32 141)(23 152 33 142)(24 153 34 143)(25 154 35 144)(26 155 36 145)(27 156 37 146)(28 157 38 147)(29 158 39 148)(30 159 40 149)(41 132 51 122)(42 133 52 123)(43 134 53 124)(44 135 54 125)(45 136 55 126)(46 137 56 127)(47 138 57 128)(48 139 58 129)(49 140 59 130)(50 121 60 131)(61 99 71 89)(62 100 72 90)(63 81 73 91)(64 82 74 92)(65 83 75 93)(66 84 76 94)(67 85 77 95)(68 86 78 96)(69 87 79 97)(70 88 80 98)
(1 121 34 84)(2 122 35 85)(3 123 36 86)(4 124 37 87)(5 125 38 88)(6 126 39 89)(7 127 40 90)(8 128 21 91)(9 129 22 92)(10 130 23 93)(11 131 24 94)(12 132 25 95)(13 133 26 96)(14 134 27 97)(15 135 28 98)(16 136 29 99)(17 137 30 100)(18 138 31 81)(19 139 32 82)(20 140 33 83)(41 154 77 116)(42 155 78 117)(43 156 79 118)(44 157 80 119)(45 158 61 120)(46 159 62 101)(47 160 63 102)(48 141 64 103)(49 142 65 104)(50 143 66 105)(51 144 67 106)(52 145 68 107)(53 146 69 108)(54 147 70 109)(55 148 71 110)(56 149 72 111)(57 150 73 112)(58 151 74 113)(59 152 75 114)(60 153 76 115)
(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 34 23)(2 22 35 9)(3 8 36 21)(4 40 37 7)(5 6 38 39)(11 20 24 33)(12 32 25 19)(13 18 26 31)(14 30 27 17)(15 16 28 29)(41 82 77 139)(42 138 78 81)(43 100 79 137)(44 136 80 99)(45 98 61 135)(46 134 62 97)(47 96 63 133)(48 132 64 95)(49 94 65 131)(50 130 66 93)(51 92 67 129)(52 128 68 91)(53 90 69 127)(54 126 70 89)(55 88 71 125)(56 124 72 87)(57 86 73 123)(58 122 74 85)(59 84 75 121)(60 140 76 83)(101 108 159 146)(102 145 160 107)(103 106 141 144)(104 143 142 105)(109 120 147 158)(110 157 148 119)(111 118 149 156)(112 155 150 117)(113 116 151 154)(114 153 152 115)
G:=sub<Sym(160)| (1,105,11,115)(2,106,12,116)(3,107,13,117)(4,108,14,118)(5,109,15,119)(6,110,16,120)(7,111,17,101)(8,112,18,102)(9,113,19,103)(10,114,20,104)(21,150,31,160)(22,151,32,141)(23,152,33,142)(24,153,34,143)(25,154,35,144)(26,155,36,145)(27,156,37,146)(28,157,38,147)(29,158,39,148)(30,159,40,149)(41,132,51,122)(42,133,52,123)(43,134,53,124)(44,135,54,125)(45,136,55,126)(46,137,56,127)(47,138,57,128)(48,139,58,129)(49,140,59,130)(50,121,60,131)(61,99,71,89)(62,100,72,90)(63,81,73,91)(64,82,74,92)(65,83,75,93)(66,84,76,94)(67,85,77,95)(68,86,78,96)(69,87,79,97)(70,88,80,98), (1,121,34,84)(2,122,35,85)(3,123,36,86)(4,124,37,87)(5,125,38,88)(6,126,39,89)(7,127,40,90)(8,128,21,91)(9,129,22,92)(10,130,23,93)(11,131,24,94)(12,132,25,95)(13,133,26,96)(14,134,27,97)(15,135,28,98)(16,136,29,99)(17,137,30,100)(18,138,31,81)(19,139,32,82)(20,140,33,83)(41,154,77,116)(42,155,78,117)(43,156,79,118)(44,157,80,119)(45,158,61,120)(46,159,62,101)(47,160,63,102)(48,141,64,103)(49,142,65,104)(50,143,66,105)(51,144,67,106)(52,145,68,107)(53,146,69,108)(54,147,70,109)(55,148,71,110)(56,149,72,111)(57,150,73,112)(58,151,74,113)(59,152,75,114)(60,153,76,115), (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), (1,10,34,23)(2,22,35,9)(3,8,36,21)(4,40,37,7)(5,6,38,39)(11,20,24,33)(12,32,25,19)(13,18,26,31)(14,30,27,17)(15,16,28,29)(41,82,77,139)(42,138,78,81)(43,100,79,137)(44,136,80,99)(45,98,61,135)(46,134,62,97)(47,96,63,133)(48,132,64,95)(49,94,65,131)(50,130,66,93)(51,92,67,129)(52,128,68,91)(53,90,69,127)(54,126,70,89)(55,88,71,125)(56,124,72,87)(57,86,73,123)(58,122,74,85)(59,84,75,121)(60,140,76,83)(101,108,159,146)(102,145,160,107)(103,106,141,144)(104,143,142,105)(109,120,147,158)(110,157,148,119)(111,118,149,156)(112,155,150,117)(113,116,151,154)(114,153,152,115)>;
G:=Group( (1,105,11,115)(2,106,12,116)(3,107,13,117)(4,108,14,118)(5,109,15,119)(6,110,16,120)(7,111,17,101)(8,112,18,102)(9,113,19,103)(10,114,20,104)(21,150,31,160)(22,151,32,141)(23,152,33,142)(24,153,34,143)(25,154,35,144)(26,155,36,145)(27,156,37,146)(28,157,38,147)(29,158,39,148)(30,159,40,149)(41,132,51,122)(42,133,52,123)(43,134,53,124)(44,135,54,125)(45,136,55,126)(46,137,56,127)(47,138,57,128)(48,139,58,129)(49,140,59,130)(50,121,60,131)(61,99,71,89)(62,100,72,90)(63,81,73,91)(64,82,74,92)(65,83,75,93)(66,84,76,94)(67,85,77,95)(68,86,78,96)(69,87,79,97)(70,88,80,98), (1,121,34,84)(2,122,35,85)(3,123,36,86)(4,124,37,87)(5,125,38,88)(6,126,39,89)(7,127,40,90)(8,128,21,91)(9,129,22,92)(10,130,23,93)(11,131,24,94)(12,132,25,95)(13,133,26,96)(14,134,27,97)(15,135,28,98)(16,136,29,99)(17,137,30,100)(18,138,31,81)(19,139,32,82)(20,140,33,83)(41,154,77,116)(42,155,78,117)(43,156,79,118)(44,157,80,119)(45,158,61,120)(46,159,62,101)(47,160,63,102)(48,141,64,103)(49,142,65,104)(50,143,66,105)(51,144,67,106)(52,145,68,107)(53,146,69,108)(54,147,70,109)(55,148,71,110)(56,149,72,111)(57,150,73,112)(58,151,74,113)(59,152,75,114)(60,153,76,115), (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), (1,10,34,23)(2,22,35,9)(3,8,36,21)(4,40,37,7)(5,6,38,39)(11,20,24,33)(12,32,25,19)(13,18,26,31)(14,30,27,17)(15,16,28,29)(41,82,77,139)(42,138,78,81)(43,100,79,137)(44,136,80,99)(45,98,61,135)(46,134,62,97)(47,96,63,133)(48,132,64,95)(49,94,65,131)(50,130,66,93)(51,92,67,129)(52,128,68,91)(53,90,69,127)(54,126,70,89)(55,88,71,125)(56,124,72,87)(57,86,73,123)(58,122,74,85)(59,84,75,121)(60,140,76,83)(101,108,159,146)(102,145,160,107)(103,106,141,144)(104,143,142,105)(109,120,147,158)(110,157,148,119)(111,118,149,156)(112,155,150,117)(113,116,151,154)(114,153,152,115) );
G=PermutationGroup([[(1,105,11,115),(2,106,12,116),(3,107,13,117),(4,108,14,118),(5,109,15,119),(6,110,16,120),(7,111,17,101),(8,112,18,102),(9,113,19,103),(10,114,20,104),(21,150,31,160),(22,151,32,141),(23,152,33,142),(24,153,34,143),(25,154,35,144),(26,155,36,145),(27,156,37,146),(28,157,38,147),(29,158,39,148),(30,159,40,149),(41,132,51,122),(42,133,52,123),(43,134,53,124),(44,135,54,125),(45,136,55,126),(46,137,56,127),(47,138,57,128),(48,139,58,129),(49,140,59,130),(50,121,60,131),(61,99,71,89),(62,100,72,90),(63,81,73,91),(64,82,74,92),(65,83,75,93),(66,84,76,94),(67,85,77,95),(68,86,78,96),(69,87,79,97),(70,88,80,98)], [(1,121,34,84),(2,122,35,85),(3,123,36,86),(4,124,37,87),(5,125,38,88),(6,126,39,89),(7,127,40,90),(8,128,21,91),(9,129,22,92),(10,130,23,93),(11,131,24,94),(12,132,25,95),(13,133,26,96),(14,134,27,97),(15,135,28,98),(16,136,29,99),(17,137,30,100),(18,138,31,81),(19,139,32,82),(20,140,33,83),(41,154,77,116),(42,155,78,117),(43,156,79,118),(44,157,80,119),(45,158,61,120),(46,159,62,101),(47,160,63,102),(48,141,64,103),(49,142,65,104),(50,143,66,105),(51,144,67,106),(52,145,68,107),(53,146,69,108),(54,147,70,109),(55,148,71,110),(56,149,72,111),(57,150,73,112),(58,151,74,113),(59,152,75,114),(60,153,76,115)], [(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)], [(1,10,34,23),(2,22,35,9),(3,8,36,21),(4,40,37,7),(5,6,38,39),(11,20,24,33),(12,32,25,19),(13,18,26,31),(14,30,27,17),(15,16,28,29),(41,82,77,139),(42,138,78,81),(43,100,79,137),(44,136,80,99),(45,98,61,135),(46,134,62,97),(47,96,63,133),(48,132,64,95),(49,94,65,131),(50,130,66,93),(51,92,67,129),(52,128,68,91),(53,90,69,127),(54,126,70,89),(55,88,71,125),(56,124,72,87),(57,86,73,123),(58,122,74,85),(59,84,75,121),(60,140,76,83),(101,108,159,146),(102,145,160,107),(103,106,141,144),(104,143,142,105),(109,120,147,158),(110,157,148,119),(111,118,149,156),(112,155,150,117),(113,116,151,154),(114,153,152,115)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | C4○D20 | C8⋊C22 | D4⋊D10 | D4.8D10 |
| kernel | C4⋊C4.236D10 | C10.D8 | C20.Q8 | D20⋊6C4 | C20.55D4 | C20⋊7D4 | C5×C42⋊C2 | C2×C20 | C22×C10 | C42⋊C2 | C20 | C4⋊C4 | C22×C4 | C10 | C2×C4 | C23 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C4⋊C4.236D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 2 | 28 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 15 | 26 |
| 23 | 21 | 0 | 0 |
| 20 | 20 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 23 | 35 | 0 | 0 |
| 20 | 18 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[2,13,0,0,28,39,0,0,0,0,15,15,0,0,15,26],[23,20,0,0,21,20,0,0,0,0,9,0,0,0,0,9],[23,20,0,0,35,18,0,0,0,0,9,0,0,0,0,32] >;
C4⋊C4.236D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{236}D_{10} % in TeX
G:=Group("C4:C4.236D10"); // GroupNames label
G:=SmallGroup(320,630);
// by ID
G=gap.SmallGroup(320,630);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*b^2*c^9>;
// generators/relations