metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.232D10, (C4×D5)⋊6Q8, C20⋊Q8⋊48C2, (C4×Q8)⋊11D5, C4.59(Q8×D5), (Q8×C20)⋊13C2, D10.1(C2×Q8), C4⋊C4.297D10, C20.117(C2×Q8), Dic5.2(C2×Q8), (D5×C42).6C2, (C4×Dic10)⋊39C2, C4.47(C4○D20), D10⋊Q8.5C2, (C2×Q8).177D10, C42⋊D5.4C2, Dic5⋊Q8⋊33C2, C20.117(C4○D4), C10.30(C22×Q8), (C2×C20).499C23, (C2×C10).122C24, (C4×C20).174C22, D10⋊2Q8.15C2, D10⋊3Q8.15C2, Dic5.10(C4○D4), Dic5.Q8⋊46C2, C4⋊Dic5.307C22, (Q8×C10).222C22, C22.143(C23×D5), C5⋊3(C23.37C23), (C2×Dic5).226C23, (C4×Dic5).284C22, (C22×D5).189C23, D10⋊C4.102C22, (C2×Dic10).299C22, C10.D4.155C22, C2.13(C2×Q8×D5), C2.30(D5×C4○D4), C10.54(C2×C4○D4), C2.61(C2×C4○D20), (C2×C4×D5).378C22, (C5×C4⋊C4).350C22, (C2×C4).584(C22×D5), SmallGroup(320,1250)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.232D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >
Subgroups: 670 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.37C23, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, Q8×C10, C4×Dic10, C4×Dic10, D5×C42, C42⋊D5, C20⋊Q8, Dic5.Q8, D10⋊Q8, D10⋊2Q8, Dic5⋊Q8, D10⋊3Q8, Q8×C20, C42.232D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, C22×D5, C23.37C23, C4○D20, Q8×D5, C23×D5, C2×C4○D20, C2×Q8×D5, D5×C4○D4, C42.232D10
►On 160 points
Generators in S160
(1 117 11 107)(2 108 12 118)(3 119 13 109)(4 110 14 120)(5 101 15 111)(6 112 16 102)(7 103 17 113)(8 114 18 104)(9 105 19 115)(10 116 20 106)(21 140 31 130)(22 131 32 121)(23 122 33 132)(24 133 34 123)(25 124 35 134)(26 135 36 125)(27 126 37 136)(28 137 38 127)(29 128 39 138)(30 139 40 129)(41 98 51 88)(42 89 52 99)(43 100 53 90)(44 91 54 81)(45 82 55 92)(46 93 56 83)(47 84 57 94)(48 95 58 85)(49 86 59 96)(50 97 60 87)(61 142 71 152)(62 153 72 143)(63 144 73 154)(64 155 74 145)(65 146 75 156)(66 157 76 147)(67 148 77 158)(68 159 78 149)(69 150 79 160)(70 141 80 151)
(1 136 78 99)(2 137 79 100)(3 138 80 81)(4 139 61 82)(5 140 62 83)(6 121 63 84)(7 122 64 85)(8 123 65 86)(9 124 66 87)(10 125 67 88)(11 126 68 89)(12 127 69 90)(13 128 70 91)(14 129 71 92)(15 130 72 93)(16 131 73 94)(17 132 74 95)(18 133 75 96)(19 134 76 97)(20 135 77 98)(21 143 56 111)(22 144 57 112)(23 145 58 113)(24 146 59 114)(25 147 60 115)(26 148 41 116)(27 149 42 117)(28 150 43 118)(29 151 44 119)(30 152 45 120)(31 153 46 101)(32 154 47 102)(33 155 48 103)(34 156 49 104)(35 157 50 105)(36 158 51 106)(37 159 52 107)(38 160 53 108)(39 141 54 109)(40 142 55 110)
(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 68 77)(2 76 69 9)(3 8 70 75)(4 74 71 7)(5 6 72 73)(11 20 78 67)(12 66 79 19)(13 18 80 65)(14 64 61 17)(15 16 62 63)(21 22 46 47)(23 40 48 45)(24 44 49 39)(25 38 50 43)(26 42 51 37)(27 36 52 41)(28 60 53 35)(29 34 54 59)(30 58 55 33)(31 32 56 57)(81 86 128 133)(82 132 129 85)(83 84 130 131)(87 100 134 127)(88 126 135 99)(89 98 136 125)(90 124 137 97)(91 96 138 123)(92 122 139 95)(93 94 140 121)(101 102 143 144)(103 120 145 142)(104 141 146 119)(105 118 147 160)(106 159 148 117)(107 116 149 158)(108 157 150 115)(109 114 151 156)(110 155 152 113)(111 112 153 154)
G:=sub<Sym(160)| (1,117,11,107)(2,108,12,118)(3,119,13,109)(4,110,14,120)(5,101,15,111)(6,112,16,102)(7,103,17,113)(8,114,18,104)(9,105,19,115)(10,116,20,106)(21,140,31,130)(22,131,32,121)(23,122,33,132)(24,133,34,123)(25,124,35,134)(26,135,36,125)(27,126,37,136)(28,137,38,127)(29,128,39,138)(30,139,40,129)(41,98,51,88)(42,89,52,99)(43,100,53,90)(44,91,54,81)(45,82,55,92)(46,93,56,83)(47,84,57,94)(48,95,58,85)(49,86,59,96)(50,97,60,87)(61,142,71,152)(62,153,72,143)(63,144,73,154)(64,155,74,145)(65,146,75,156)(66,157,76,147)(67,148,77,158)(68,159,78,149)(69,150,79,160)(70,141,80,151), (1,136,78,99)(2,137,79,100)(3,138,80,81)(4,139,61,82)(5,140,62,83)(6,121,63,84)(7,122,64,85)(8,123,65,86)(9,124,66,87)(10,125,67,88)(11,126,68,89)(12,127,69,90)(13,128,70,91)(14,129,71,92)(15,130,72,93)(16,131,73,94)(17,132,74,95)(18,133,75,96)(19,134,76,97)(20,135,77,98)(21,143,56,111)(22,144,57,112)(23,145,58,113)(24,146,59,114)(25,147,60,115)(26,148,41,116)(27,149,42,117)(28,150,43,118)(29,151,44,119)(30,152,45,120)(31,153,46,101)(32,154,47,102)(33,155,48,103)(34,156,49,104)(35,157,50,105)(36,158,51,106)(37,159,52,107)(38,160,53,108)(39,141,54,109)(40,142,55,110), (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), (1,10,68,77)(2,76,69,9)(3,8,70,75)(4,74,71,7)(5,6,72,73)(11,20,78,67)(12,66,79,19)(13,18,80,65)(14,64,61,17)(15,16,62,63)(21,22,46,47)(23,40,48,45)(24,44,49,39)(25,38,50,43)(26,42,51,37)(27,36,52,41)(28,60,53,35)(29,34,54,59)(30,58,55,33)(31,32,56,57)(81,86,128,133)(82,132,129,85)(83,84,130,131)(87,100,134,127)(88,126,135,99)(89,98,136,125)(90,124,137,97)(91,96,138,123)(92,122,139,95)(93,94,140,121)(101,102,143,144)(103,120,145,142)(104,141,146,119)(105,118,147,160)(106,159,148,117)(107,116,149,158)(108,157,150,115)(109,114,151,156)(110,155,152,113)(111,112,153,154)>;
G:=Group( (1,117,11,107)(2,108,12,118)(3,119,13,109)(4,110,14,120)(5,101,15,111)(6,112,16,102)(7,103,17,113)(8,114,18,104)(9,105,19,115)(10,116,20,106)(21,140,31,130)(22,131,32,121)(23,122,33,132)(24,133,34,123)(25,124,35,134)(26,135,36,125)(27,126,37,136)(28,137,38,127)(29,128,39,138)(30,139,40,129)(41,98,51,88)(42,89,52,99)(43,100,53,90)(44,91,54,81)(45,82,55,92)(46,93,56,83)(47,84,57,94)(48,95,58,85)(49,86,59,96)(50,97,60,87)(61,142,71,152)(62,153,72,143)(63,144,73,154)(64,155,74,145)(65,146,75,156)(66,157,76,147)(67,148,77,158)(68,159,78,149)(69,150,79,160)(70,141,80,151), (1,136,78,99)(2,137,79,100)(3,138,80,81)(4,139,61,82)(5,140,62,83)(6,121,63,84)(7,122,64,85)(8,123,65,86)(9,124,66,87)(10,125,67,88)(11,126,68,89)(12,127,69,90)(13,128,70,91)(14,129,71,92)(15,130,72,93)(16,131,73,94)(17,132,74,95)(18,133,75,96)(19,134,76,97)(20,135,77,98)(21,143,56,111)(22,144,57,112)(23,145,58,113)(24,146,59,114)(25,147,60,115)(26,148,41,116)(27,149,42,117)(28,150,43,118)(29,151,44,119)(30,152,45,120)(31,153,46,101)(32,154,47,102)(33,155,48,103)(34,156,49,104)(35,157,50,105)(36,158,51,106)(37,159,52,107)(38,160,53,108)(39,141,54,109)(40,142,55,110), (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), (1,10,68,77)(2,76,69,9)(3,8,70,75)(4,74,71,7)(5,6,72,73)(11,20,78,67)(12,66,79,19)(13,18,80,65)(14,64,61,17)(15,16,62,63)(21,22,46,47)(23,40,48,45)(24,44,49,39)(25,38,50,43)(26,42,51,37)(27,36,52,41)(28,60,53,35)(29,34,54,59)(30,58,55,33)(31,32,56,57)(81,86,128,133)(82,132,129,85)(83,84,130,131)(87,100,134,127)(88,126,135,99)(89,98,136,125)(90,124,137,97)(91,96,138,123)(92,122,139,95)(93,94,140,121)(101,102,143,144)(103,120,145,142)(104,141,146,119)(105,118,147,160)(106,159,148,117)(107,116,149,158)(108,157,150,115)(109,114,151,156)(110,155,152,113)(111,112,153,154) );
G=PermutationGroup([[(1,117,11,107),(2,108,12,118),(3,119,13,109),(4,110,14,120),(5,101,15,111),(6,112,16,102),(7,103,17,113),(8,114,18,104),(9,105,19,115),(10,116,20,106),(21,140,31,130),(22,131,32,121),(23,122,33,132),(24,133,34,123),(25,124,35,134),(26,135,36,125),(27,126,37,136),(28,137,38,127),(29,128,39,138),(30,139,40,129),(41,98,51,88),(42,89,52,99),(43,100,53,90),(44,91,54,81),(45,82,55,92),(46,93,56,83),(47,84,57,94),(48,95,58,85),(49,86,59,96),(50,97,60,87),(61,142,71,152),(62,153,72,143),(63,144,73,154),(64,155,74,145),(65,146,75,156),(66,157,76,147),(67,148,77,158),(68,159,78,149),(69,150,79,160),(70,141,80,151)], [(1,136,78,99),(2,137,79,100),(3,138,80,81),(4,139,61,82),(5,140,62,83),(6,121,63,84),(7,122,64,85),(8,123,65,86),(9,124,66,87),(10,125,67,88),(11,126,68,89),(12,127,69,90),(13,128,70,91),(14,129,71,92),(15,130,72,93),(16,131,73,94),(17,132,74,95),(18,133,75,96),(19,134,76,97),(20,135,77,98),(21,143,56,111),(22,144,57,112),(23,145,58,113),(24,146,59,114),(25,147,60,115),(26,148,41,116),(27,149,42,117),(28,150,43,118),(29,151,44,119),(30,152,45,120),(31,153,46,101),(32,154,47,102),(33,155,48,103),(34,156,49,104),(35,157,50,105),(36,158,51,106),(37,159,52,107),(38,160,53,108),(39,141,54,109),(40,142,55,110)], [(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)], [(1,10,68,77),(2,76,69,9),(3,8,70,75),(4,74,71,7),(5,6,72,73),(11,20,78,67),(12,66,79,19),(13,18,80,65),(14,64,61,17),(15,16,62,63),(21,22,46,47),(23,40,48,45),(24,44,49,39),(25,38,50,43),(26,42,51,37),(27,36,52,41),(28,60,53,35),(29,34,54,59),(30,58,55,33),(31,32,56,57),(81,86,128,133),(82,132,129,85),(83,84,130,131),(87,100,134,127),(88,126,135,99),(89,98,136,125),(90,124,137,97),(91,96,138,123),(92,122,139,95),(93,94,140,121),(101,102,143,144),(103,120,145,142),(104,141,146,119),(105,118,147,160),(106,159,148,117),(107,116,149,158),(108,157,150,115),(109,114,151,156),(110,155,152,113),(111,112,153,154)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | C4○D20 | Q8×D5 | D5×C4○D4 |
| kernel | C42.232D10 | C4×Dic10 | D5×C42 | C42⋊D5 | C20⋊Q8 | Dic5.Q8 | D10⋊Q8 | D10⋊2Q8 | Dic5⋊Q8 | D10⋊3Q8 | Q8×C20 | C4×D5 | C4×Q8 | Dic5 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 6 | 2 | 16 | 4 | 4 |
Matrix representation of C42.232D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 28 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 3 | 38 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 29 | 37 |
| 0 | 0 | 26 | 12 |
| 24 | 3 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 12 | 4 |
| 0 | 0 | 36 | 29 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,28,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[3,3,0,0,38,24,0,0,0,0,29,26,0,0,37,12],[24,40,0,0,3,17,0,0,0,0,12,36,0,0,4,29] >;
C42.232D10 in GAP, Magma, Sage, TeX
C_4^2._{232}D_{10} % in TeX
G:=Group("C4^2.232D10"); // GroupNames label
G:=SmallGroup(320,1250);
// by ID
G=gap.SmallGroup(320,1250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^9>;
// generators/relations