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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4, C2×D4 [×2], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C5⋊2C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C22×D5, C22×C10, C23.19D4, C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×D20, C2×C5⋊D4, C22×C20, C10.D8, C20.Q8, D20⋊6C4 [×2], C20.55D4, C20⋊7D4, C5×C42⋊C2, C4⋊C4.236D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, C23.19D4, C4○D20 [×2], C2×C5⋊D4, C23.23D10, D4⋊D10, D4.8D10, C4⋊C4.236D10
►On 160 points
Generators in S160
(1 28 11 38)(2 29 12 39)(3 30 13 40)(4 31 14 21)(5 32 15 22)(6 33 16 23)(7 34 17 24)(8 35 18 25)(9 36 19 26)(10 37 20 27)(41 135 51 125)(42 136 52 126)(43 137 53 127)(44 138 54 128)(45 139 55 129)(46 140 56 130)(47 121 57 131)(48 122 58 132)(49 123 59 133)(50 124 60 134)(61 103 71 113)(62 104 72 114)(63 105 73 115)(64 106 74 116)(65 107 75 117)(66 108 76 118)(67 109 77 119)(68 110 78 120)(69 111 79 101)(70 112 80 102)(81 152 91 142)(82 153 92 143)(83 154 93 144)(84 155 94 145)(85 156 95 146)(86 157 96 147)(87 158 97 148)(88 159 98 149)(89 160 99 150)(90 141 100 151)
(1 98 52 65)(2 99 53 66)(3 100 54 67)(4 81 55 68)(5 82 56 69)(6 83 57 70)(7 84 58 71)(8 85 59 72)(9 86 60 73)(10 87 41 74)(11 88 42 75)(12 89 43 76)(13 90 44 77)(14 91 45 78)(15 92 46 79)(16 93 47 80)(17 94 48 61)(18 95 49 62)(19 96 50 63)(20 97 51 64)(21 152 139 110)(22 153 140 111)(23 154 121 112)(24 155 122 113)(25 156 123 114)(26 157 124 115)(27 158 125 116)(28 159 126 117)(29 160 127 118)(30 141 128 119)(31 142 129 120)(32 143 130 101)(33 144 131 102)(34 145 132 103)(35 146 133 104)(36 147 134 105)(37 148 135 106)(38 149 136 107)(39 150 137 108)(40 151 138 109)
(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 52 41)(2 60 53 9)(3 8 54 59)(4 58 55 7)(5 6 56 57)(11 20 42 51)(12 50 43 19)(13 18 44 49)(14 48 45 17)(15 16 46 47)(21 132 139 34)(22 33 140 131)(23 130 121 32)(24 31 122 129)(25 128 123 30)(26 29 124 127)(27 126 125 28)(35 138 133 40)(36 39 134 137)(37 136 135 38)(61 110 94 152)(62 151 95 109)(63 108 96 150)(64 149 97 107)(65 106 98 148)(66 147 99 105)(67 104 100 146)(68 145 81 103)(69 102 82 144)(70 143 83 101)(71 120 84 142)(72 141 85 119)(73 118 86 160)(74 159 87 117)(75 116 88 158)(76 157 89 115)(77 114 90 156)(78 155 91 113)(79 112 92 154)(80 153 93 111)
G:=sub<Sym(160)| (1,28,11,38)(2,29,12,39)(3,30,13,40)(4,31,14,21)(5,32,15,22)(6,33,16,23)(7,34,17,24)(8,35,18,25)(9,36,19,26)(10,37,20,27)(41,135,51,125)(42,136,52,126)(43,137,53,127)(44,138,54,128)(45,139,55,129)(46,140,56,130)(47,121,57,131)(48,122,58,132)(49,123,59,133)(50,124,60,134)(61,103,71,113)(62,104,72,114)(63,105,73,115)(64,106,74,116)(65,107,75,117)(66,108,76,118)(67,109,77,119)(68,110,78,120)(69,111,79,101)(70,112,80,102)(81,152,91,142)(82,153,92,143)(83,154,93,144)(84,155,94,145)(85,156,95,146)(86,157,96,147)(87,158,97,148)(88,159,98,149)(89,160,99,150)(90,141,100,151), (1,98,52,65)(2,99,53,66)(3,100,54,67)(4,81,55,68)(5,82,56,69)(6,83,57,70)(7,84,58,71)(8,85,59,72)(9,86,60,73)(10,87,41,74)(11,88,42,75)(12,89,43,76)(13,90,44,77)(14,91,45,78)(15,92,46,79)(16,93,47,80)(17,94,48,61)(18,95,49,62)(19,96,50,63)(20,97,51,64)(21,152,139,110)(22,153,140,111)(23,154,121,112)(24,155,122,113)(25,156,123,114)(26,157,124,115)(27,158,125,116)(28,159,126,117)(29,160,127,118)(30,141,128,119)(31,142,129,120)(32,143,130,101)(33,144,131,102)(34,145,132,103)(35,146,133,104)(36,147,134,105)(37,148,135,106)(38,149,136,107)(39,150,137,108)(40,151,138,109), (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), (1,10,52,41)(2,60,53,9)(3,8,54,59)(4,58,55,7)(5,6,56,57)(11,20,42,51)(12,50,43,19)(13,18,44,49)(14,48,45,17)(15,16,46,47)(21,132,139,34)(22,33,140,131)(23,130,121,32)(24,31,122,129)(25,128,123,30)(26,29,124,127)(27,126,125,28)(35,138,133,40)(36,39,134,137)(37,136,135,38)(61,110,94,152)(62,151,95,109)(63,108,96,150)(64,149,97,107)(65,106,98,148)(66,147,99,105)(67,104,100,146)(68,145,81,103)(69,102,82,144)(70,143,83,101)(71,120,84,142)(72,141,85,119)(73,118,86,160)(74,159,87,117)(75,116,88,158)(76,157,89,115)(77,114,90,156)(78,155,91,113)(79,112,92,154)(80,153,93,111)>;
G:=Group( (1,28,11,38)(2,29,12,39)(3,30,13,40)(4,31,14,21)(5,32,15,22)(6,33,16,23)(7,34,17,24)(8,35,18,25)(9,36,19,26)(10,37,20,27)(41,135,51,125)(42,136,52,126)(43,137,53,127)(44,138,54,128)(45,139,55,129)(46,140,56,130)(47,121,57,131)(48,122,58,132)(49,123,59,133)(50,124,60,134)(61,103,71,113)(62,104,72,114)(63,105,73,115)(64,106,74,116)(65,107,75,117)(66,108,76,118)(67,109,77,119)(68,110,78,120)(69,111,79,101)(70,112,80,102)(81,152,91,142)(82,153,92,143)(83,154,93,144)(84,155,94,145)(85,156,95,146)(86,157,96,147)(87,158,97,148)(88,159,98,149)(89,160,99,150)(90,141,100,151), (1,98,52,65)(2,99,53,66)(3,100,54,67)(4,81,55,68)(5,82,56,69)(6,83,57,70)(7,84,58,71)(8,85,59,72)(9,86,60,73)(10,87,41,74)(11,88,42,75)(12,89,43,76)(13,90,44,77)(14,91,45,78)(15,92,46,79)(16,93,47,80)(17,94,48,61)(18,95,49,62)(19,96,50,63)(20,97,51,64)(21,152,139,110)(22,153,140,111)(23,154,121,112)(24,155,122,113)(25,156,123,114)(26,157,124,115)(27,158,125,116)(28,159,126,117)(29,160,127,118)(30,141,128,119)(31,142,129,120)(32,143,130,101)(33,144,131,102)(34,145,132,103)(35,146,133,104)(36,147,134,105)(37,148,135,106)(38,149,136,107)(39,150,137,108)(40,151,138,109), (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), (1,10,52,41)(2,60,53,9)(3,8,54,59)(4,58,55,7)(5,6,56,57)(11,20,42,51)(12,50,43,19)(13,18,44,49)(14,48,45,17)(15,16,46,47)(21,132,139,34)(22,33,140,131)(23,130,121,32)(24,31,122,129)(25,128,123,30)(26,29,124,127)(27,126,125,28)(35,138,133,40)(36,39,134,137)(37,136,135,38)(61,110,94,152)(62,151,95,109)(63,108,96,150)(64,149,97,107)(65,106,98,148)(66,147,99,105)(67,104,100,146)(68,145,81,103)(69,102,82,144)(70,143,83,101)(71,120,84,142)(72,141,85,119)(73,118,86,160)(74,159,87,117)(75,116,88,158)(76,157,89,115)(77,114,90,156)(78,155,91,113)(79,112,92,154)(80,153,93,111) );
G=PermutationGroup([(1,28,11,38),(2,29,12,39),(3,30,13,40),(4,31,14,21),(5,32,15,22),(6,33,16,23),(7,34,17,24),(8,35,18,25),(9,36,19,26),(10,37,20,27),(41,135,51,125),(42,136,52,126),(43,137,53,127),(44,138,54,128),(45,139,55,129),(46,140,56,130),(47,121,57,131),(48,122,58,132),(49,123,59,133),(50,124,60,134),(61,103,71,113),(62,104,72,114),(63,105,73,115),(64,106,74,116),(65,107,75,117),(66,108,76,118),(67,109,77,119),(68,110,78,120),(69,111,79,101),(70,112,80,102),(81,152,91,142),(82,153,92,143),(83,154,93,144),(84,155,94,145),(85,156,95,146),(86,157,96,147),(87,158,97,148),(88,159,98,149),(89,160,99,150),(90,141,100,151)], [(1,98,52,65),(2,99,53,66),(3,100,54,67),(4,81,55,68),(5,82,56,69),(6,83,57,70),(7,84,58,71),(8,85,59,72),(9,86,60,73),(10,87,41,74),(11,88,42,75),(12,89,43,76),(13,90,44,77),(14,91,45,78),(15,92,46,79),(16,93,47,80),(17,94,48,61),(18,95,49,62),(19,96,50,63),(20,97,51,64),(21,152,139,110),(22,153,140,111),(23,154,121,112),(24,155,122,113),(25,156,123,114),(26,157,124,115),(27,158,125,116),(28,159,126,117),(29,160,127,118),(30,141,128,119),(31,142,129,120),(32,143,130,101),(33,144,131,102),(34,145,132,103),(35,146,133,104),(36,147,134,105),(37,148,135,106),(38,149,136,107),(39,150,137,108),(40,151,138,109)], [(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)], [(1,10,52,41),(2,60,53,9),(3,8,54,59),(4,58,55,7),(5,6,56,57),(11,20,42,51),(12,50,43,19),(13,18,44,49),(14,48,45,17),(15,16,46,47),(21,132,139,34),(22,33,140,131),(23,130,121,32),(24,31,122,129),(25,128,123,30),(26,29,124,127),(27,126,125,28),(35,138,133,40),(36,39,134,137),(37,136,135,38),(61,110,94,152),(62,151,95,109),(63,108,96,150),(64,149,97,107),(65,106,98,148),(66,147,99,105),(67,104,100,146),(68,145,81,103),(69,102,82,144),(70,143,83,101),(71,120,84,142),(72,141,85,119),(73,118,86,160),(74,159,87,117),(75,116,88,158),(76,157,89,115),(77,114,90,156),(78,155,91,113),(79,112,92,154),(80,153,93,111)])
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