metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.230D10, (C2×C20).285D4, (C2×C10).17Q16, C10.33(C2×Q16), C4.88(C4○D20), C10.Q16⋊25C2, C10.D8⋊26C2, (C22×C4).96D10, C20.176(C4○D4), C10.85(C8⋊C22), (C2×C20).323C23, C20.48D4.9C2, C20.55D4.3C2, (C22×C10).188D4, C23.79(C5⋊D4), C5⋊4(C23.48D4), C22.5(C5⋊Q16), C2.7(D4.D10), C4⋊Dic5.132C22, (C22×C20).138C22, (C2×Dic10).100C22, C10.60(C22.D4), C2.10(C23.23D10), (C2×C4⋊C4).8D5, (C10×C4⋊C4).7C2, C2.5(C2×C5⋊Q16), (C2×C10).443(C2×D4), (C2×C4).33(C5⋊D4), (C5×C4⋊C4).261C22, (C2×C5⋊2C8).83C22, (C2×C4).423(C22×D5), C22.132(C2×C5⋊D4), SmallGroup(320,597)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.230D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >
Subgroups: 318 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.48D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, C22×C20, C10.D8, C10.Q16, C20.55D4, C20.48D4, C10×C4⋊C4, C4⋊C4.230D10
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C22.D4, C2×Q16, C8⋊C22, C5⋊D4, C22×D5, C23.48D4, C5⋊Q16, C4○D20, C2×C5⋊D4, C23.23D10, D4.D10, C2×C5⋊Q16, C4⋊C4.230D10
►On 160 points
Generators in S160
(1 34 8 37)(2 35 9 38)(3 31 10 39)(4 32 6 40)(5 33 7 36)(11 21 16 29)(12 22 17 30)(13 23 18 26)(14 24 19 27)(15 25 20 28)(41 78 49 75)(42 79 50 71)(43 80 46 72)(44 76 47 73)(45 77 48 74)(51 68 60 63)(52 69 56 64)(53 70 57 65)(54 66 58 61)(55 67 59 62)(81 114 106 129)(82 115 107 130)(83 116 108 121)(84 117 109 122)(85 118 110 123)(86 119 101 124)(87 120 102 125)(88 111 103 126)(89 112 104 127)(90 113 105 128)(91 131 156 146)(92 132 157 147)(93 133 158 148)(94 134 159 149)(95 135 160 150)(96 136 151 141)(97 137 152 142)(98 138 153 143)(99 139 154 144)(100 140 155 145)
(1 58 18 44)(2 59 19 45)(3 60 20 41)(4 56 16 42)(5 57 17 43)(6 52 11 50)(7 53 12 46)(8 54 13 47)(9 55 14 48)(10 51 15 49)(21 79 40 64)(22 80 36 65)(23 76 37 61)(24 77 38 62)(25 78 39 63)(26 73 34 66)(27 74 35 67)(28 75 31 68)(29 71 32 69)(30 72 33 70)(81 159 101 99)(82 160 102 100)(83 151 103 91)(84 152 104 92)(85 153 105 93)(86 154 106 94)(87 155 107 95)(88 156 108 96)(89 157 109 97)(90 158 110 98)(111 131 121 141)(112 132 122 142)(113 133 123 143)(114 134 124 144)(115 135 125 145)(116 136 126 146)(117 137 127 147)(118 138 128 148)(119 139 129 149)(120 140 130 150)
(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 81 18 101)(2 85 19 105)(3 89 20 109)(4 83 16 103)(5 87 17 107)(6 108 11 88)(7 102 12 82)(8 106 13 86)(9 110 14 90)(10 104 15 84)(21 126 40 116)(22 130 36 120)(23 124 37 114)(24 128 38 118)(25 122 39 112)(26 119 34 129)(27 113 35 123)(28 117 31 127)(29 111 32 121)(30 115 33 125)(41 142 60 132)(42 146 56 136)(43 150 57 140)(44 144 58 134)(45 148 59 138)(46 135 53 145)(47 139 54 149)(48 133 55 143)(49 137 51 147)(50 131 52 141)(61 94 76 154)(62 98 77 158)(63 92 78 152)(64 96 79 156)(65 100 80 160)(66 159 73 99)(67 153 74 93)(68 157 75 97)(69 151 71 91)(70 155 72 95)
G:=sub<Sym(160)| (1,34,8,37)(2,35,9,38)(3,31,10,39)(4,32,6,40)(5,33,7,36)(11,21,16,29)(12,22,17,30)(13,23,18,26)(14,24,19,27)(15,25,20,28)(41,78,49,75)(42,79,50,71)(43,80,46,72)(44,76,47,73)(45,77,48,74)(51,68,60,63)(52,69,56,64)(53,70,57,65)(54,66,58,61)(55,67,59,62)(81,114,106,129)(82,115,107,130)(83,116,108,121)(84,117,109,122)(85,118,110,123)(86,119,101,124)(87,120,102,125)(88,111,103,126)(89,112,104,127)(90,113,105,128)(91,131,156,146)(92,132,157,147)(93,133,158,148)(94,134,159,149)(95,135,160,150)(96,136,151,141)(97,137,152,142)(98,138,153,143)(99,139,154,144)(100,140,155,145), (1,58,18,44)(2,59,19,45)(3,60,20,41)(4,56,16,42)(5,57,17,43)(6,52,11,50)(7,53,12,46)(8,54,13,47)(9,55,14,48)(10,51,15,49)(21,79,40,64)(22,80,36,65)(23,76,37,61)(24,77,38,62)(25,78,39,63)(26,73,34,66)(27,74,35,67)(28,75,31,68)(29,71,32,69)(30,72,33,70)(81,159,101,99)(82,160,102,100)(83,151,103,91)(84,152,104,92)(85,153,105,93)(86,154,106,94)(87,155,107,95)(88,156,108,96)(89,157,109,97)(90,158,110,98)(111,131,121,141)(112,132,122,142)(113,133,123,143)(114,134,124,144)(115,135,125,145)(116,136,126,146)(117,137,127,147)(118,138,128,148)(119,139,129,149)(120,140,130,150), (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), (1,81,18,101)(2,85,19,105)(3,89,20,109)(4,83,16,103)(5,87,17,107)(6,108,11,88)(7,102,12,82)(8,106,13,86)(9,110,14,90)(10,104,15,84)(21,126,40,116)(22,130,36,120)(23,124,37,114)(24,128,38,118)(25,122,39,112)(26,119,34,129)(27,113,35,123)(28,117,31,127)(29,111,32,121)(30,115,33,125)(41,142,60,132)(42,146,56,136)(43,150,57,140)(44,144,58,134)(45,148,59,138)(46,135,53,145)(47,139,54,149)(48,133,55,143)(49,137,51,147)(50,131,52,141)(61,94,76,154)(62,98,77,158)(63,92,78,152)(64,96,79,156)(65,100,80,160)(66,159,73,99)(67,153,74,93)(68,157,75,97)(69,151,71,91)(70,155,72,95)>;
G:=Group( (1,34,8,37)(2,35,9,38)(3,31,10,39)(4,32,6,40)(5,33,7,36)(11,21,16,29)(12,22,17,30)(13,23,18,26)(14,24,19,27)(15,25,20,28)(41,78,49,75)(42,79,50,71)(43,80,46,72)(44,76,47,73)(45,77,48,74)(51,68,60,63)(52,69,56,64)(53,70,57,65)(54,66,58,61)(55,67,59,62)(81,114,106,129)(82,115,107,130)(83,116,108,121)(84,117,109,122)(85,118,110,123)(86,119,101,124)(87,120,102,125)(88,111,103,126)(89,112,104,127)(90,113,105,128)(91,131,156,146)(92,132,157,147)(93,133,158,148)(94,134,159,149)(95,135,160,150)(96,136,151,141)(97,137,152,142)(98,138,153,143)(99,139,154,144)(100,140,155,145), (1,58,18,44)(2,59,19,45)(3,60,20,41)(4,56,16,42)(5,57,17,43)(6,52,11,50)(7,53,12,46)(8,54,13,47)(9,55,14,48)(10,51,15,49)(21,79,40,64)(22,80,36,65)(23,76,37,61)(24,77,38,62)(25,78,39,63)(26,73,34,66)(27,74,35,67)(28,75,31,68)(29,71,32,69)(30,72,33,70)(81,159,101,99)(82,160,102,100)(83,151,103,91)(84,152,104,92)(85,153,105,93)(86,154,106,94)(87,155,107,95)(88,156,108,96)(89,157,109,97)(90,158,110,98)(111,131,121,141)(112,132,122,142)(113,133,123,143)(114,134,124,144)(115,135,125,145)(116,136,126,146)(117,137,127,147)(118,138,128,148)(119,139,129,149)(120,140,130,150), (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), (1,81,18,101)(2,85,19,105)(3,89,20,109)(4,83,16,103)(5,87,17,107)(6,108,11,88)(7,102,12,82)(8,106,13,86)(9,110,14,90)(10,104,15,84)(21,126,40,116)(22,130,36,120)(23,124,37,114)(24,128,38,118)(25,122,39,112)(26,119,34,129)(27,113,35,123)(28,117,31,127)(29,111,32,121)(30,115,33,125)(41,142,60,132)(42,146,56,136)(43,150,57,140)(44,144,58,134)(45,148,59,138)(46,135,53,145)(47,139,54,149)(48,133,55,143)(49,137,51,147)(50,131,52,141)(61,94,76,154)(62,98,77,158)(63,92,78,152)(64,96,79,156)(65,100,80,160)(66,159,73,99)(67,153,74,93)(68,157,75,97)(69,151,71,91)(70,155,72,95) );
G=PermutationGroup([[(1,34,8,37),(2,35,9,38),(3,31,10,39),(4,32,6,40),(5,33,7,36),(11,21,16,29),(12,22,17,30),(13,23,18,26),(14,24,19,27),(15,25,20,28),(41,78,49,75),(42,79,50,71),(43,80,46,72),(44,76,47,73),(45,77,48,74),(51,68,60,63),(52,69,56,64),(53,70,57,65),(54,66,58,61),(55,67,59,62),(81,114,106,129),(82,115,107,130),(83,116,108,121),(84,117,109,122),(85,118,110,123),(86,119,101,124),(87,120,102,125),(88,111,103,126),(89,112,104,127),(90,113,105,128),(91,131,156,146),(92,132,157,147),(93,133,158,148),(94,134,159,149),(95,135,160,150),(96,136,151,141),(97,137,152,142),(98,138,153,143),(99,139,154,144),(100,140,155,145)], [(1,58,18,44),(2,59,19,45),(3,60,20,41),(4,56,16,42),(5,57,17,43),(6,52,11,50),(7,53,12,46),(8,54,13,47),(9,55,14,48),(10,51,15,49),(21,79,40,64),(22,80,36,65),(23,76,37,61),(24,77,38,62),(25,78,39,63),(26,73,34,66),(27,74,35,67),(28,75,31,68),(29,71,32,69),(30,72,33,70),(81,159,101,99),(82,160,102,100),(83,151,103,91),(84,152,104,92),(85,153,105,93),(86,154,106,94),(87,155,107,95),(88,156,108,96),(89,157,109,97),(90,158,110,98),(111,131,121,141),(112,132,122,142),(113,133,123,143),(114,134,124,144),(115,135,125,145),(116,136,126,146),(117,137,127,147),(118,138,128,148),(119,139,129,149),(120,140,130,150)], [(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)], [(1,81,18,101),(2,85,19,105),(3,89,20,109),(4,83,16,103),(5,87,17,107),(6,108,11,88),(7,102,12,82),(8,106,13,86),(9,110,14,90),(10,104,15,84),(21,126,40,116),(22,130,36,120),(23,124,37,114),(24,128,38,118),(25,122,39,112),(26,119,34,129),(27,113,35,123),(28,117,31,127),(29,111,32,121),(30,115,33,125),(41,142,60,132),(42,146,56,136),(43,150,57,140),(44,144,58,134),(45,148,59,138),(46,135,53,145),(47,139,54,149),(48,133,55,143),(49,137,51,147),(50,131,52,141),(61,94,76,154),(62,98,77,158),(63,92,78,152),(64,96,79,156),(65,100,80,160),(66,159,73,99),(67,153,74,93),(68,157,75,97),(69,151,71,91),(70,155,72,95)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 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 | D4 | D4 | D5 | C4○D4 | Q16 | D10 | D10 | C5⋊D4 | C5⋊D4 | C4○D20 | C8⋊C22 | C5⋊Q16 | D4.D10 |
| kernel | C4⋊C4.230D10 | C10.D8 | C10.Q16 | C20.55D4 | C20.48D4 | C10×C4⋊C4 | C2×C20 | C22×C10 | C2×C4⋊C4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C23 | C4 | C10 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of C4⋊C4.230D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 14 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 5 | 24 |
| 0 | 0 | 16 | 36 |
| 10 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 37 | 0 | 0 |
| 31 | 0 | 0 | 0 |
| 0 | 0 | 26 | 10 |
| 0 | 0 | 2 | 15 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,14,0,0,0,9],[32,0,0,0,0,9,0,0,0,0,5,16,0,0,24,36],[10,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,31,0,0,37,0,0,0,0,0,26,2,0,0,10,15] >;
C4⋊C4.230D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{230}D_{10} % in TeX
G:=Group("C4:C4.230D10"); // GroupNames label
G:=SmallGroup(320,597);
// by ID
G=gap.SmallGroup(320,597);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,268,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,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^-1>;
// generators/relations