metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.231D10, (C2×C20).286D4, C4.89(C4○D20), C10.Q16⋊26C2, C20.Q8⋊26C2, C10.50(C2×SD16), (C2×C10).33SD16, (C22×C4).97D10, C20.177(C4○D4), (C2×C20).324C23, C20.55D4.4C2, (C22×C10).189D4, C23.80(C5⋊D4), C5⋊4(C23.47D4), C22.8(D4.D5), C20.48D4.10C2, C2.7(C20.C23), C10.85(C8.C22), C4⋊Dic5.133C22, (C22×C20).139C22, (C2×Dic10).101C22, C10.61(C22.D4), C2.11(C23.23D10), (C2×C4⋊C4).9D5, (C10×C4⋊C4).8C2, C2.5(C2×D4.D5), (C2×C10).444(C2×D4), (C2×C4).34(C5⋊D4), (C5×C4⋊C4).262C22, (C2×C5⋊2C8).84C22, (C2×C4).424(C22×D5), C22.133(C2×C5⋊D4), SmallGroup(320,598)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.231D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=b2c-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, C4.Q8, C2×C4⋊C4, C22⋊Q8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.47D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, C22×C20, C20.Q8, C10.Q16, C20.55D4, C20.48D4, C10×C4⋊C4, C4⋊C4.231D10
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8.C22, C5⋊D4, C22×D5, C23.47D4, D4.D5, C4○D20, C2×C5⋊D4, C23.23D10, C2×D4.D5, C20.C23, C4⋊C4.231D10
►On 160 points
Generators in S160
(1 36 8 33)(2 37 9 34)(3 38 10 35)(4 39 6 31)(5 40 7 32)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 74 49 77)(42 75 50 78)(43 71 46 79)(44 72 47 80)(45 73 48 76)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)(81 123 134 160)(82 124 135 151)(83 125 136 152)(84 126 137 153)(85 127 138 154)(86 128 139 155)(87 129 140 156)(88 130 131 157)(89 121 132 158)(90 122 133 159)(91 143 120 110)(92 144 111 101)(93 145 112 102)(94 146 113 103)(95 147 114 104)(96 148 115 105)(97 149 116 106)(98 150 117 107)(99 141 118 108)(100 142 119 109)
(1 51 13 43)(2 52 14 44)(3 53 15 45)(4 54 11 41)(5 55 12 42)(6 59 16 49)(7 60 17 50)(8 56 18 46)(9 57 19 47)(10 58 20 48)(21 74 31 64)(22 75 32 65)(23 71 33 61)(24 72 34 62)(25 73 35 63)(26 77 39 69)(27 78 40 70)(28 79 36 66)(29 80 37 67)(30 76 38 68)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 129 106 124)(102 130 107 125)(103 121 108 126)(104 122 109 127)(105 123 110 128)(111 140 116 135)(112 131 117 136)(113 132 118 137)(114 133 119 138)(115 134 120 139)(141 153 146 158)(142 154 147 159)(143 155 148 160)(144 156 149 151)(145 157 150 152)
(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)
(1 104 18 142)(2 108 19 146)(3 102 20 150)(4 106 16 144)(5 110 17 148)(6 149 11 101)(7 143 12 105)(8 147 13 109)(9 141 14 103)(10 145 15 107)(21 92 39 116)(22 96 40 120)(23 100 36 114)(24 94 37 118)(25 98 38 112)(26 111 31 97)(27 115 32 91)(28 119 33 95)(29 113 34 99)(30 117 35 93)(41 140 59 82)(42 134 60 86)(43 138 56 90)(44 132 57 84)(45 136 58 88)(46 85 51 133)(47 89 52 137)(48 83 53 131)(49 87 54 135)(50 81 55 139)(61 122 79 154)(62 126 80 158)(63 130 76 152)(64 124 77 156)(65 128 78 160)(66 159 71 127)(67 153 72 121)(68 157 73 125)(69 151 74 129)(70 155 75 123)
G:=sub<Sym(160)| (1,36,8,33)(2,37,9,34)(3,38,10,35)(4,39,6,31)(5,40,7,32)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25)(41,74,49,77)(42,75,50,78)(43,71,46,79)(44,72,47,80)(45,73,48,76)(51,61,56,66)(52,62,57,67)(53,63,58,68)(54,64,59,69)(55,65,60,70)(81,123,134,160)(82,124,135,151)(83,125,136,152)(84,126,137,153)(85,127,138,154)(86,128,139,155)(87,129,140,156)(88,130,131,157)(89,121,132,158)(90,122,133,159)(91,143,120,110)(92,144,111,101)(93,145,112,102)(94,146,113,103)(95,147,114,104)(96,148,115,105)(97,149,116,106)(98,150,117,107)(99,141,118,108)(100,142,119,109), (1,51,13,43)(2,52,14,44)(3,53,15,45)(4,54,11,41)(5,55,12,42)(6,59,16,49)(7,60,17,50)(8,56,18,46)(9,57,19,47)(10,58,20,48)(21,74,31,64)(22,75,32,65)(23,71,33,61)(24,72,34,62)(25,73,35,63)(26,77,39,69)(27,78,40,70)(28,79,36,66)(29,80,37,67)(30,76,38,68)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,129,106,124)(102,130,107,125)(103,121,108,126)(104,122,109,127)(105,123,110,128)(111,140,116,135)(112,131,117,136)(113,132,118,137)(114,133,119,138)(115,134,120,139)(141,153,146,158)(142,154,147,159)(143,155,148,160)(144,156,149,151)(145,157,150,152), (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), (1,104,18,142)(2,108,19,146)(3,102,20,150)(4,106,16,144)(5,110,17,148)(6,149,11,101)(7,143,12,105)(8,147,13,109)(9,141,14,103)(10,145,15,107)(21,92,39,116)(22,96,40,120)(23,100,36,114)(24,94,37,118)(25,98,38,112)(26,111,31,97)(27,115,32,91)(28,119,33,95)(29,113,34,99)(30,117,35,93)(41,140,59,82)(42,134,60,86)(43,138,56,90)(44,132,57,84)(45,136,58,88)(46,85,51,133)(47,89,52,137)(48,83,53,131)(49,87,54,135)(50,81,55,139)(61,122,79,154)(62,126,80,158)(63,130,76,152)(64,124,77,156)(65,128,78,160)(66,159,71,127)(67,153,72,121)(68,157,73,125)(69,151,74,129)(70,155,75,123)>;
G:=Group( (1,36,8,33)(2,37,9,34)(3,38,10,35)(4,39,6,31)(5,40,7,32)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25)(41,74,49,77)(42,75,50,78)(43,71,46,79)(44,72,47,80)(45,73,48,76)(51,61,56,66)(52,62,57,67)(53,63,58,68)(54,64,59,69)(55,65,60,70)(81,123,134,160)(82,124,135,151)(83,125,136,152)(84,126,137,153)(85,127,138,154)(86,128,139,155)(87,129,140,156)(88,130,131,157)(89,121,132,158)(90,122,133,159)(91,143,120,110)(92,144,111,101)(93,145,112,102)(94,146,113,103)(95,147,114,104)(96,148,115,105)(97,149,116,106)(98,150,117,107)(99,141,118,108)(100,142,119,109), (1,51,13,43)(2,52,14,44)(3,53,15,45)(4,54,11,41)(5,55,12,42)(6,59,16,49)(7,60,17,50)(8,56,18,46)(9,57,19,47)(10,58,20,48)(21,74,31,64)(22,75,32,65)(23,71,33,61)(24,72,34,62)(25,73,35,63)(26,77,39,69)(27,78,40,70)(28,79,36,66)(29,80,37,67)(30,76,38,68)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,129,106,124)(102,130,107,125)(103,121,108,126)(104,122,109,127)(105,123,110,128)(111,140,116,135)(112,131,117,136)(113,132,118,137)(114,133,119,138)(115,134,120,139)(141,153,146,158)(142,154,147,159)(143,155,148,160)(144,156,149,151)(145,157,150,152), (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), (1,104,18,142)(2,108,19,146)(3,102,20,150)(4,106,16,144)(5,110,17,148)(6,149,11,101)(7,143,12,105)(8,147,13,109)(9,141,14,103)(10,145,15,107)(21,92,39,116)(22,96,40,120)(23,100,36,114)(24,94,37,118)(25,98,38,112)(26,111,31,97)(27,115,32,91)(28,119,33,95)(29,113,34,99)(30,117,35,93)(41,140,59,82)(42,134,60,86)(43,138,56,90)(44,132,57,84)(45,136,58,88)(46,85,51,133)(47,89,52,137)(48,83,53,131)(49,87,54,135)(50,81,55,139)(61,122,79,154)(62,126,80,158)(63,130,76,152)(64,124,77,156)(65,128,78,160)(66,159,71,127)(67,153,72,121)(68,157,73,125)(69,151,74,129)(70,155,75,123) );
G=PermutationGroup([[(1,36,8,33),(2,37,9,34),(3,38,10,35),(4,39,6,31),(5,40,7,32),(11,26,16,21),(12,27,17,22),(13,28,18,23),(14,29,19,24),(15,30,20,25),(41,74,49,77),(42,75,50,78),(43,71,46,79),(44,72,47,80),(45,73,48,76),(51,61,56,66),(52,62,57,67),(53,63,58,68),(54,64,59,69),(55,65,60,70),(81,123,134,160),(82,124,135,151),(83,125,136,152),(84,126,137,153),(85,127,138,154),(86,128,139,155),(87,129,140,156),(88,130,131,157),(89,121,132,158),(90,122,133,159),(91,143,120,110),(92,144,111,101),(93,145,112,102),(94,146,113,103),(95,147,114,104),(96,148,115,105),(97,149,116,106),(98,150,117,107),(99,141,118,108),(100,142,119,109)], [(1,51,13,43),(2,52,14,44),(3,53,15,45),(4,54,11,41),(5,55,12,42),(6,59,16,49),(7,60,17,50),(8,56,18,46),(9,57,19,47),(10,58,20,48),(21,74,31,64),(22,75,32,65),(23,71,33,61),(24,72,34,62),(25,73,35,63),(26,77,39,69),(27,78,40,70),(28,79,36,66),(29,80,37,67),(30,76,38,68),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,129,106,124),(102,130,107,125),(103,121,108,126),(104,122,109,127),(105,123,110,128),(111,140,116,135),(112,131,117,136),(113,132,118,137),(114,133,119,138),(115,134,120,139),(141,153,146,158),(142,154,147,159),(143,155,148,160),(144,156,149,151),(145,157,150,152)], [(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)], [(1,104,18,142),(2,108,19,146),(3,102,20,150),(4,106,16,144),(5,110,17,148),(6,149,11,101),(7,143,12,105),(8,147,13,109),(9,141,14,103),(10,145,15,107),(21,92,39,116),(22,96,40,120),(23,100,36,114),(24,94,37,118),(25,98,38,112),(26,111,31,97),(27,115,32,91),(28,119,33,95),(29,113,34,99),(30,117,35,93),(41,140,59,82),(42,134,60,86),(43,138,56,90),(44,132,57,84),(45,136,58,88),(46,85,51,133),(47,89,52,137),(48,83,53,131),(49,87,54,135),(50,81,55,139),(61,122,79,154),(62,126,80,158),(63,130,76,152),(64,124,77,156),(65,128,78,160),(66,159,71,127),(67,153,72,121),(68,157,73,125),(69,151,74,129),(70,155,75,123)]])
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 | SD16 | D10 | D10 | C5⋊D4 | C5⋊D4 | C4○D20 | C8.C22 | D4.D5 | C20.C23 |
| kernel | C4⋊C4.231D10 | C20.Q8 | 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⋊C4.231D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 40 | 24 |
| 1 | 14 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 33 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 4 | 0 | 0 | 0 | 0 |
| 2 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 0 | 0 |
| 0 | 0 | 3 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 11 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,17,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,40,0,0,0,0,1,24],[1,0,0,0,0,0,14,40,0,0,0,0,0,0,25,33,0,0,0,0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,2,0,0,0,0,4,14,0,0,0,0,0,0,3,3,0,0,0,0,11,38,0,0,0,0,0,0,9,11,0,0,0,0,0,32] >;
C4⋊C4.231D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{231}D_{10} % in TeX
G:=Group("C4:C4.231D10"); // GroupNames label
G:=SmallGroup(320,598);
// by ID
G=gap.SmallGroup(320,598);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^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^-1*b,d*c*d^-1=b^2*c^-1>;
// generators/relations