metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.233D10, (C2×C20).449D4, C4.90(C4○D20), C10.83(C4○D8), C10.D8⋊28C2, C10.Q16⋊27C2, C20.Q8⋊28C2, (C22×C10).72D4, C42⋊C2.5D5, C20.178(C4○D4), (C2×C20).326C23, C20.55D4.5C2, (C22×C4).108D10, C5⋊6(C23.20D4), C23.19(C5⋊D4), C20.48D4.11C2, C2.6(D4.9D10), C2.8(D4.8D10), C4⋊Dic5.134C22, C10.105(C8.C22), (C22×C20).146C22, (C2×Dic10).102C22, C10.65(C22.D4), C2.16(C23.23D10), (C2×C10).455(C2×D4), (C2×C4).214(C5⋊D4), (C5×C4⋊C4).264C22, (C5×C42⋊C2).5C2, (C2×C5⋊2C8).86C22, (C2×C4).426(C22×D5), C22.144(C2×C5⋊D4), SmallGroup(320,623)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.233D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c9 >
Subgroups: 302 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.20D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, C10.D8, C20.Q8, C10.Q16, C20.55D4, C20.48D4, C5×C42⋊C2, C4⋊C4.233D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, C5⋊D4, C22×D5, C23.20D4, C4○D20, C2×C5⋊D4, C23.23D10, D4.8D10, D4.9D10, C4⋊C4.233D10
►On 160 points
Generators in S160
(1 122 11 132)(2 123 12 133)(3 124 13 134)(4 125 14 135)(5 126 15 136)(6 127 16 137)(7 128 17 138)(8 129 18 139)(9 130 19 140)(10 131 20 121)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 86 71 96)(62 87 72 97)(63 88 73 98)(64 89 74 99)(65 90 75 100)(66 91 76 81)(67 92 77 82)(68 93 78 83)(69 94 79 84)(70 95 80 85)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)
(1 85 127 65)(2 86 128 66)(3 87 129 67)(4 88 130 68)(5 89 131 69)(6 90 132 70)(7 91 133 71)(8 92 134 72)(9 93 135 73)(10 94 136 74)(11 95 137 75)(12 96 138 76)(13 97 139 77)(14 98 140 78)(15 99 121 79)(16 100 122 80)(17 81 123 61)(18 82 124 62)(19 83 125 63)(20 84 126 64)(21 144 111 44)(22 145 112 45)(23 146 113 46)(24 147 114 47)(25 148 115 48)(26 149 116 49)(27 150 117 50)(28 151 118 51)(29 152 119 52)(30 153 120 53)(31 154 101 54)(32 155 102 55)(33 156 103 56)(34 157 104 57)(35 158 105 58)(36 159 106 59)(37 160 107 60)(38 141 108 41)(39 142 109 42)(40 143 110 43)
(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 28 11 38)(2 117 12 107)(3 26 13 36)(4 115 14 105)(5 24 15 34)(6 113 16 103)(7 22 17 32)(8 111 18 101)(9 40 19 30)(10 109 20 119)(21 124 31 134)(23 122 33 132)(25 140 35 130)(27 138 37 128)(29 136 39 126)(41 100 51 90)(42 79 52 69)(43 98 53 88)(44 77 54 67)(45 96 55 86)(46 75 56 65)(47 94 57 84)(48 73 58 63)(49 92 59 82)(50 71 60 61)(62 149 72 159)(64 147 74 157)(66 145 76 155)(68 143 78 153)(70 141 80 151)(81 150 91 160)(83 148 93 158)(85 146 95 156)(87 144 97 154)(89 142 99 152)(102 133 112 123)(104 131 114 121)(106 129 116 139)(108 127 118 137)(110 125 120 135)
G:=sub<Sym(160)| (1,122,11,132)(2,123,12,133)(3,124,13,134)(4,125,14,135)(5,126,15,136)(6,127,16,137)(7,128,17,138)(8,129,18,139)(9,130,19,140)(10,131,20,121)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,86,71,96)(62,87,72,97)(63,88,73,98)(64,89,74,99)(65,90,75,100)(66,91,76,81)(67,92,77,82)(68,93,78,83)(69,94,79,84)(70,95,80,85)(101,116,111,106)(102,117,112,107)(103,118,113,108)(104,119,114,109)(105,120,115,110)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160), (1,85,127,65)(2,86,128,66)(3,87,129,67)(4,88,130,68)(5,89,131,69)(6,90,132,70)(7,91,133,71)(8,92,134,72)(9,93,135,73)(10,94,136,74)(11,95,137,75)(12,96,138,76)(13,97,139,77)(14,98,140,78)(15,99,121,79)(16,100,122,80)(17,81,123,61)(18,82,124,62)(19,83,125,63)(20,84,126,64)(21,144,111,44)(22,145,112,45)(23,146,113,46)(24,147,114,47)(25,148,115,48)(26,149,116,49)(27,150,117,50)(28,151,118,51)(29,152,119,52)(30,153,120,53)(31,154,101,54)(32,155,102,55)(33,156,103,56)(34,157,104,57)(35,158,105,58)(36,159,106,59)(37,160,107,60)(38,141,108,41)(39,142,109,42)(40,143,110,43), (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), (1,28,11,38)(2,117,12,107)(3,26,13,36)(4,115,14,105)(5,24,15,34)(6,113,16,103)(7,22,17,32)(8,111,18,101)(9,40,19,30)(10,109,20,119)(21,124,31,134)(23,122,33,132)(25,140,35,130)(27,138,37,128)(29,136,39,126)(41,100,51,90)(42,79,52,69)(43,98,53,88)(44,77,54,67)(45,96,55,86)(46,75,56,65)(47,94,57,84)(48,73,58,63)(49,92,59,82)(50,71,60,61)(62,149,72,159)(64,147,74,157)(66,145,76,155)(68,143,78,153)(70,141,80,151)(81,150,91,160)(83,148,93,158)(85,146,95,156)(87,144,97,154)(89,142,99,152)(102,133,112,123)(104,131,114,121)(106,129,116,139)(108,127,118,137)(110,125,120,135)>;
G:=Group( (1,122,11,132)(2,123,12,133)(3,124,13,134)(4,125,14,135)(5,126,15,136)(6,127,16,137)(7,128,17,138)(8,129,18,139)(9,130,19,140)(10,131,20,121)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,86,71,96)(62,87,72,97)(63,88,73,98)(64,89,74,99)(65,90,75,100)(66,91,76,81)(67,92,77,82)(68,93,78,83)(69,94,79,84)(70,95,80,85)(101,116,111,106)(102,117,112,107)(103,118,113,108)(104,119,114,109)(105,120,115,110)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160), (1,85,127,65)(2,86,128,66)(3,87,129,67)(4,88,130,68)(5,89,131,69)(6,90,132,70)(7,91,133,71)(8,92,134,72)(9,93,135,73)(10,94,136,74)(11,95,137,75)(12,96,138,76)(13,97,139,77)(14,98,140,78)(15,99,121,79)(16,100,122,80)(17,81,123,61)(18,82,124,62)(19,83,125,63)(20,84,126,64)(21,144,111,44)(22,145,112,45)(23,146,113,46)(24,147,114,47)(25,148,115,48)(26,149,116,49)(27,150,117,50)(28,151,118,51)(29,152,119,52)(30,153,120,53)(31,154,101,54)(32,155,102,55)(33,156,103,56)(34,157,104,57)(35,158,105,58)(36,159,106,59)(37,160,107,60)(38,141,108,41)(39,142,109,42)(40,143,110,43), (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), (1,28,11,38)(2,117,12,107)(3,26,13,36)(4,115,14,105)(5,24,15,34)(6,113,16,103)(7,22,17,32)(8,111,18,101)(9,40,19,30)(10,109,20,119)(21,124,31,134)(23,122,33,132)(25,140,35,130)(27,138,37,128)(29,136,39,126)(41,100,51,90)(42,79,52,69)(43,98,53,88)(44,77,54,67)(45,96,55,86)(46,75,56,65)(47,94,57,84)(48,73,58,63)(49,92,59,82)(50,71,60,61)(62,149,72,159)(64,147,74,157)(66,145,76,155)(68,143,78,153)(70,141,80,151)(81,150,91,160)(83,148,93,158)(85,146,95,156)(87,144,97,154)(89,142,99,152)(102,133,112,123)(104,131,114,121)(106,129,116,139)(108,127,118,137)(110,125,120,135) );
G=PermutationGroup([[(1,122,11,132),(2,123,12,133),(3,124,13,134),(4,125,14,135),(5,126,15,136),(6,127,16,137),(7,128,17,138),(8,129,18,139),(9,130,19,140),(10,131,20,121),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,86,71,96),(62,87,72,97),(63,88,73,98),(64,89,74,99),(65,90,75,100),(66,91,76,81),(67,92,77,82),(68,93,78,83),(69,94,79,84),(70,95,80,85),(101,116,111,106),(102,117,112,107),(103,118,113,108),(104,119,114,109),(105,120,115,110),(141,146,151,156),(142,147,152,157),(143,148,153,158),(144,149,154,159),(145,150,155,160)], [(1,85,127,65),(2,86,128,66),(3,87,129,67),(4,88,130,68),(5,89,131,69),(6,90,132,70),(7,91,133,71),(8,92,134,72),(9,93,135,73),(10,94,136,74),(11,95,137,75),(12,96,138,76),(13,97,139,77),(14,98,140,78),(15,99,121,79),(16,100,122,80),(17,81,123,61),(18,82,124,62),(19,83,125,63),(20,84,126,64),(21,144,111,44),(22,145,112,45),(23,146,113,46),(24,147,114,47),(25,148,115,48),(26,149,116,49),(27,150,117,50),(28,151,118,51),(29,152,119,52),(30,153,120,53),(31,154,101,54),(32,155,102,55),(33,156,103,56),(34,157,104,57),(35,158,105,58),(36,159,106,59),(37,160,107,60),(38,141,108,41),(39,142,109,42),(40,143,110,43)], [(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)], [(1,28,11,38),(2,117,12,107),(3,26,13,36),(4,115,14,105),(5,24,15,34),(6,113,16,103),(7,22,17,32),(8,111,18,101),(9,40,19,30),(10,109,20,119),(21,124,31,134),(23,122,33,132),(25,140,35,130),(27,138,37,128),(29,136,39,126),(41,100,51,90),(42,79,52,69),(43,98,53,88),(44,77,54,67),(45,96,55,86),(46,75,56,65),(47,94,57,84),(48,73,58,63),(49,92,59,82),(50,71,60,61),(62,149,72,159),(64,147,74,157),(66,145,76,155),(68,143,78,153),(70,141,80,151),(81,150,91,160),(83,148,93,158),(85,146,95,156),(87,144,97,154),(89,142,99,152),(102,133,112,123),(104,131,114,121),(106,129,116,139),(108,127,118,137),(110,125,120,135)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 40 | 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 | D4.8D10 | D4.9D10 |
| kernel | C4⋊C4.233D10 | C10.D8 | C20.Q8 | C10.Q16 | C20.55D4 | C20.48D4 | 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⋊C4.233D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 32 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 23 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,0,32,0,0,32,0],[25,0,0,0,0,18,0,0,0,0,32,0,0,0,0,32],[0,25,0,0,23,0,0,0,0,0,0,3,0,0,27,0] >;
C4⋊C4.233D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{233}D_{10} % in TeX
G:=Group("C4:C4.233D10"); // GroupNames label
G:=SmallGroup(320,623);
// by ID
G=gap.SmallGroup(320,623);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,232,254,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^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