metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.3Q8, C4.3Dic10, C4⋊C4.6D5, C10.6(C2×Q8), (C2×C4).43D10, C4⋊Dic5.7C2, C5⋊3(C42.C2), (C4×Dic5).2C2, C2.8(C2×Dic10), C10.25(C4○D4), (C2×C20).22C22, (C2×C10).31C23, C2.4(Q8⋊2D5), C10.D4.3C2, C2.12(D4⋊2D5), C22.48(C22×D5), (C2×Dic5).10C22, (C5×C4⋊C4).7C2, SmallGroup(160,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic10
G = < a,b,c | a4=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a2b-1 >
Subgroups: 144 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, C10, C42, C4⋊C4, C4⋊C4, Dic5, C20, C20, C2×C10, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C4.Dic10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, D10, C42.C2, Dic10, C22×D5, C2×Dic10, D4⋊2D5, Q8⋊2D5, C4.Dic10
►Regular action on 160 points
(1 37 80 129)(2 130 61 38)(3 39 62 131)(4 132 63 40)(5 21 64 133)(6 134 65 22)(7 23 66 135)(8 136 67 24)(9 25 68 137)(10 138 69 26)(11 27 70 139)(12 140 71 28)(13 29 72 121)(14 122 73 30)(15 31 74 123)(16 124 75 32)(17 33 76 125)(18 126 77 34)(19 35 78 127)(20 128 79 36)(41 95 145 115)(42 116 146 96)(43 97 147 117)(44 118 148 98)(45 99 149 119)(46 120 150 100)(47 81 151 101)(48 102 152 82)(49 83 153 103)(50 104 154 84)(51 85 155 105)(52 106 156 86)(53 87 157 107)(54 108 158 88)(55 89 159 109)(56 110 160 90)(57 91 141 111)(58 112 142 92)(59 93 143 113)(60 114 144 94)
(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)
(1 147 11 157)(2 42 12 52)(3 145 13 155)(4 60 14 50)(5 143 15 153)(6 58 16 48)(7 141 17 151)(8 56 18 46)(9 159 19 149)(10 54 20 44)(21 113 31 103)(22 92 32 82)(23 111 33 101)(24 90 34 100)(25 109 35 119)(26 88 36 98)(27 107 37 117)(28 86 38 96)(29 105 39 115)(30 84 40 94)(41 72 51 62)(43 70 53 80)(45 68 55 78)(47 66 57 76)(49 64 59 74)(61 146 71 156)(63 144 73 154)(65 142 75 152)(67 160 77 150)(69 158 79 148)(81 135 91 125)(83 133 93 123)(85 131 95 121)(87 129 97 139)(89 127 99 137)(102 134 112 124)(104 132 114 122)(106 130 116 140)(108 128 118 138)(110 126 120 136)
G:=sub<Sym(160)| (1,37,80,129)(2,130,61,38)(3,39,62,131)(4,132,63,40)(5,21,64,133)(6,134,65,22)(7,23,66,135)(8,136,67,24)(9,25,68,137)(10,138,69,26)(11,27,70,139)(12,140,71,28)(13,29,72,121)(14,122,73,30)(15,31,74,123)(16,124,75,32)(17,33,76,125)(18,126,77,34)(19,35,78,127)(20,128,79,36)(41,95,145,115)(42,116,146,96)(43,97,147,117)(44,118,148,98)(45,99,149,119)(46,120,150,100)(47,81,151,101)(48,102,152,82)(49,83,153,103)(50,104,154,84)(51,85,155,105)(52,106,156,86)(53,87,157,107)(54,108,158,88)(55,89,159,109)(56,110,160,90)(57,91,141,111)(58,112,142,92)(59,93,143,113)(60,114,144,94), (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), (1,147,11,157)(2,42,12,52)(3,145,13,155)(4,60,14,50)(5,143,15,153)(6,58,16,48)(7,141,17,151)(8,56,18,46)(9,159,19,149)(10,54,20,44)(21,113,31,103)(22,92,32,82)(23,111,33,101)(24,90,34,100)(25,109,35,119)(26,88,36,98)(27,107,37,117)(28,86,38,96)(29,105,39,115)(30,84,40,94)(41,72,51,62)(43,70,53,80)(45,68,55,78)(47,66,57,76)(49,64,59,74)(61,146,71,156)(63,144,73,154)(65,142,75,152)(67,160,77,150)(69,158,79,148)(81,135,91,125)(83,133,93,123)(85,131,95,121)(87,129,97,139)(89,127,99,137)(102,134,112,124)(104,132,114,122)(106,130,116,140)(108,128,118,138)(110,126,120,136)>;
G:=Group( (1,37,80,129)(2,130,61,38)(3,39,62,131)(4,132,63,40)(5,21,64,133)(6,134,65,22)(7,23,66,135)(8,136,67,24)(9,25,68,137)(10,138,69,26)(11,27,70,139)(12,140,71,28)(13,29,72,121)(14,122,73,30)(15,31,74,123)(16,124,75,32)(17,33,76,125)(18,126,77,34)(19,35,78,127)(20,128,79,36)(41,95,145,115)(42,116,146,96)(43,97,147,117)(44,118,148,98)(45,99,149,119)(46,120,150,100)(47,81,151,101)(48,102,152,82)(49,83,153,103)(50,104,154,84)(51,85,155,105)(52,106,156,86)(53,87,157,107)(54,108,158,88)(55,89,159,109)(56,110,160,90)(57,91,141,111)(58,112,142,92)(59,93,143,113)(60,114,144,94), (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), (1,147,11,157)(2,42,12,52)(3,145,13,155)(4,60,14,50)(5,143,15,153)(6,58,16,48)(7,141,17,151)(8,56,18,46)(9,159,19,149)(10,54,20,44)(21,113,31,103)(22,92,32,82)(23,111,33,101)(24,90,34,100)(25,109,35,119)(26,88,36,98)(27,107,37,117)(28,86,38,96)(29,105,39,115)(30,84,40,94)(41,72,51,62)(43,70,53,80)(45,68,55,78)(47,66,57,76)(49,64,59,74)(61,146,71,156)(63,144,73,154)(65,142,75,152)(67,160,77,150)(69,158,79,148)(81,135,91,125)(83,133,93,123)(85,131,95,121)(87,129,97,139)(89,127,99,137)(102,134,112,124)(104,132,114,122)(106,130,116,140)(108,128,118,138)(110,126,120,136) );
G=PermutationGroup([[(1,37,80,129),(2,130,61,38),(3,39,62,131),(4,132,63,40),(5,21,64,133),(6,134,65,22),(7,23,66,135),(8,136,67,24),(9,25,68,137),(10,138,69,26),(11,27,70,139),(12,140,71,28),(13,29,72,121),(14,122,73,30),(15,31,74,123),(16,124,75,32),(17,33,76,125),(18,126,77,34),(19,35,78,127),(20,128,79,36),(41,95,145,115),(42,116,146,96),(43,97,147,117),(44,118,148,98),(45,99,149,119),(46,120,150,100),(47,81,151,101),(48,102,152,82),(49,83,153,103),(50,104,154,84),(51,85,155,105),(52,106,156,86),(53,87,157,107),(54,108,158,88),(55,89,159,109),(56,110,160,90),(57,91,141,111),(58,112,142,92),(59,93,143,113),(60,114,144,94)], [(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)], [(1,147,11,157),(2,42,12,52),(3,145,13,155),(4,60,14,50),(5,143,15,153),(6,58,16,48),(7,141,17,151),(8,56,18,46),(9,159,19,149),(10,54,20,44),(21,113,31,103),(22,92,32,82),(23,111,33,101),(24,90,34,100),(25,109,35,119),(26,88,36,98),(27,107,37,117),(28,86,38,96),(29,105,39,115),(30,84,40,94),(41,72,51,62),(43,70,53,80),(45,68,55,78),(47,66,57,76),(49,64,59,74),(61,146,71,156),(63,144,73,154),(65,142,75,152),(67,160,77,150),(69,158,79,148),(81,135,91,125),(83,133,93,123),(85,131,95,121),(87,129,97,139),(89,127,99,137),(102,134,112,124),(104,132,114,122),(106,130,116,140),(108,128,118,138),(110,126,120,136)]])
C4.Dic10 is a maximal subgroup of
C10.C4≀C2 D4.Dic10 C4⋊C4.D10 D4.2Dic10 (C8×Dic5)⋊C2 Q8.Dic10 C40⋊8C4.C2 Q8.2Dic10 Q8⋊Dic5⋊C2 C40⋊3Q8 Dic10.Q8 C8.8Dic10 D20.Q8 C40⋊4Q8 Dic10.2Q8 C8.6Dic10 D20.2Q8 C10.12- 1+4 C10.52- 1+4 C10.112+ 1+4 C42.88D10 C42.90D10 C42.94D10 C42.95D10 D4⋊5Dic10 D4⋊6Dic10 C42.229D10 C42.116D10 Q8⋊5Dic10 Q8⋊6Dic10 C42.131D10 C42.134D10 C4⋊C4.178D10 C10.432+ 1+4 C10.1152+ 1+4 C10.472+ 1+4 C22⋊Q8⋊25D5 C10.152- 1+4 C10.212- 1+4 C10.772- 1+4 C10.572+ 1+4 C10.582+ 1+4 C10.802- 1+4 C10.632+ 1+4 C10.852- 1+4 C10.692+ 1+4 Dic10⋊7Q8 C42.147D10 D5×C42.C2 C42.148D10 D20⋊7Q8 C42.152D10 C42.154D10 C42.155D10 C42.159D10 C42.161D10 C42.162D10 C42.165D10 Dic10⋊8Q8 C42.241D10 C42.174D10 D20⋊9Q8 Dic3.Dic10 Dic3.2Dic10 C12.Dic10 C20.Dic6 C4.Dic30
C4.Dic10 is a maximal quotient of
C4⋊Dic5⋊15C4 C10.52(C4×D4) (C2×C4).Dic10 C10.(C4⋊Q8) C20⋊4(C4⋊C4) C20.48(C4⋊C4) (C2×C20).54D4 C20⋊6(C4⋊C4) (C2×C20).55D4 Dic3.Dic10 Dic3.2Dic10 C12.Dic10 C20.Dic6 C4.Dic30
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | Dic10 | D4⋊2D5 | Q8⋊2D5 |
| kernel | C4.Dic10 | C4×Dic5 | C10.D4 | C4⋊Dic5 | C5×C4⋊C4 | C20 | C4⋊C4 | C10 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 2 | 2 | 4 | 6 | 8 | 2 | 2 |
Matrix representation of C4.Dic10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 3 |
| 0 | 0 | 3 | 21 |
| 11 | 2 | 0 | 0 |
| 25 | 27 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 18 | 27 | 0 | 0 |
| 32 | 23 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,20,3,0,0,3,21],[11,25,0,0,2,27,0,0,0,0,0,40,0,0,1,0],[18,32,0,0,27,23,0,0,0,0,32,0,0,0,0,32] >;
C4.Dic10 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{10} % in TeX
G:=Group("C4.Dic10"); // GroupNames label
G:=SmallGroup(160,111);
// by ID
G=gap.SmallGroup(160,111);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,506,188,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=1,c^2=b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations