metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.3D12, C15⋊(C4.D4), (C22×S3).F5, C5⋊(C12.46D4), C22.F5⋊1S3, C3⋊1(C23.F5), C22.5(S3×F5), C2.14(D6⋊F5), C10.14(D6⋊C4), C15⋊8M4(2)⋊1C2, (C2×Dic5).71D6, (C3×Dic5).32D4, (C22×D15).4C4, C6.14(C22⋊F5), C30.14(C22⋊C4), Dic5.3(C3⋊D4), (C6×Dic5).138C22, (S3×C2×C10).3C4, (C2×C6).3(C2×F5), (C2×C30).8(C2×C4), (C2×C10).10(C4×S3), (C3×C22.F5)⋊1C2, (C2×C5⋊D12).10C2, SmallGroup(480,250)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.D12
G = < a,b,c,d | a10=1, b2=c12=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, cbc-1=a5b, bd=db, dcd-1=a5bc11 >
Subgroups: 660 in 92 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, M4(2), C2×D4, Dic5, D10, C2×C10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C30, C4.D4, C5⋊C8, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C4.Dic3, C3×M4(2), C2×D12, C3×Dic5, S3×C10, D30, C2×C30, C22.F5, C22.F5, C2×C5⋊D4, C12.46D4, C3×C5⋊C8, C15⋊C8, C5⋊D12, C6×Dic5, S3×C2×C10, C22×D15, C23.F5, C3×C22.F5, C15⋊8M4(2), C2×C5⋊D12, Dic5.D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C4.D4, C2×F5, D6⋊C4, C22⋊F5, C12.46D4, S3×F5, C23.F5, D6⋊F5, Dic5.D12
►On 120 points
Generators in S120
(1 98 58 32 75 13 110 70 44 87)(2 33 111 88 59 14 45 99 76 71)(3 89 46 72 112 15 77 34 60 100)(4 49 78 101 47 16 61 90 113 35)(5 102 62 36 79 17 114 50 48 91)(6 37 115 92 63 18 25 103 80 51)(7 93 26 52 116 19 81 38 64 104)(8 53 82 105 27 20 65 94 117 39)(9 106 66 40 83 21 118 54 28 95)(10 41 119 96 67 22 29 107 84 55)(11 73 30 56 120 23 85 42 68 108)(12 57 86 109 31 24 69 74 97 43)
(1 19 13 7)(2 8 14 20)(3 21 15 9)(4 10 16 22)(5 23 17 11)(6 12 18 24)(25 31 37 43)(26 44 38 32)(27 33 39 45)(28 46 40 34)(29 35 41 47)(30 48 42 36)(49 55 61 67)(50 68 62 56)(51 57 63 69)(52 70 64 58)(53 59 65 71)(54 72 66 60)(73 91 85 79)(74 80 86 92)(75 93 87 81)(76 82 88 94)(77 95 89 83)(78 84 90 96)(97 103 109 115)(98 116 110 104)(99 105 111 117)(100 118 112 106)(101 107 113 119)(102 120 114 108)
(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)
(1 6 19 12 13 18 7 24)(2 23 8 17 14 11 20 5)(3 4 21 10 15 16 9 22)(25 38 31 32 37 26 43 44)(27 36 33 30 39 48 45 42)(28 41 46 47 40 29 34 35)(49 66 55 60 61 54 67 72)(50 71 68 53 62 59 56 65)(51 64 57 58 63 52 69 70)(73 82 91 88 85 94 79 76)(74 75 80 93 86 87 92 81)(77 78 95 84 89 90 83 96)(97 98 103 116 109 110 115 104)(99 120 105 114 111 108 117 102)(100 101 118 107 112 113 106 119)
G:=sub<Sym(120)| (1,98,58,32,75,13,110,70,44,87)(2,33,111,88,59,14,45,99,76,71)(3,89,46,72,112,15,77,34,60,100)(4,49,78,101,47,16,61,90,113,35)(5,102,62,36,79,17,114,50,48,91)(6,37,115,92,63,18,25,103,80,51)(7,93,26,52,116,19,81,38,64,104)(8,53,82,105,27,20,65,94,117,39)(9,106,66,40,83,21,118,54,28,95)(10,41,119,96,67,22,29,107,84,55)(11,73,30,56,120,23,85,42,68,108)(12,57,86,109,31,24,69,74,97,43), (1,19,13,7)(2,8,14,20)(3,21,15,9)(4,10,16,22)(5,23,17,11)(6,12,18,24)(25,31,37,43)(26,44,38,32)(27,33,39,45)(28,46,40,34)(29,35,41,47)(30,48,42,36)(49,55,61,67)(50,68,62,56)(51,57,63,69)(52,70,64,58)(53,59,65,71)(54,72,66,60)(73,91,85,79)(74,80,86,92)(75,93,87,81)(76,82,88,94)(77,95,89,83)(78,84,90,96)(97,103,109,115)(98,116,110,104)(99,105,111,117)(100,118,112,106)(101,107,113,119)(102,120,114,108), (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), (1,6,19,12,13,18,7,24)(2,23,8,17,14,11,20,5)(3,4,21,10,15,16,9,22)(25,38,31,32,37,26,43,44)(27,36,33,30,39,48,45,42)(28,41,46,47,40,29,34,35)(49,66,55,60,61,54,67,72)(50,71,68,53,62,59,56,65)(51,64,57,58,63,52,69,70)(73,82,91,88,85,94,79,76)(74,75,80,93,86,87,92,81)(77,78,95,84,89,90,83,96)(97,98,103,116,109,110,115,104)(99,120,105,114,111,108,117,102)(100,101,118,107,112,113,106,119)>;
G:=Group( (1,98,58,32,75,13,110,70,44,87)(2,33,111,88,59,14,45,99,76,71)(3,89,46,72,112,15,77,34,60,100)(4,49,78,101,47,16,61,90,113,35)(5,102,62,36,79,17,114,50,48,91)(6,37,115,92,63,18,25,103,80,51)(7,93,26,52,116,19,81,38,64,104)(8,53,82,105,27,20,65,94,117,39)(9,106,66,40,83,21,118,54,28,95)(10,41,119,96,67,22,29,107,84,55)(11,73,30,56,120,23,85,42,68,108)(12,57,86,109,31,24,69,74,97,43), (1,19,13,7)(2,8,14,20)(3,21,15,9)(4,10,16,22)(5,23,17,11)(6,12,18,24)(25,31,37,43)(26,44,38,32)(27,33,39,45)(28,46,40,34)(29,35,41,47)(30,48,42,36)(49,55,61,67)(50,68,62,56)(51,57,63,69)(52,70,64,58)(53,59,65,71)(54,72,66,60)(73,91,85,79)(74,80,86,92)(75,93,87,81)(76,82,88,94)(77,95,89,83)(78,84,90,96)(97,103,109,115)(98,116,110,104)(99,105,111,117)(100,118,112,106)(101,107,113,119)(102,120,114,108), (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), (1,6,19,12,13,18,7,24)(2,23,8,17,14,11,20,5)(3,4,21,10,15,16,9,22)(25,38,31,32,37,26,43,44)(27,36,33,30,39,48,45,42)(28,41,46,47,40,29,34,35)(49,66,55,60,61,54,67,72)(50,71,68,53,62,59,56,65)(51,64,57,58,63,52,69,70)(73,82,91,88,85,94,79,76)(74,75,80,93,86,87,92,81)(77,78,95,84,89,90,83,96)(97,98,103,116,109,110,115,104)(99,120,105,114,111,108,117,102)(100,101,118,107,112,113,106,119) );
G=PermutationGroup([[(1,98,58,32,75,13,110,70,44,87),(2,33,111,88,59,14,45,99,76,71),(3,89,46,72,112,15,77,34,60,100),(4,49,78,101,47,16,61,90,113,35),(5,102,62,36,79,17,114,50,48,91),(6,37,115,92,63,18,25,103,80,51),(7,93,26,52,116,19,81,38,64,104),(8,53,82,105,27,20,65,94,117,39),(9,106,66,40,83,21,118,54,28,95),(10,41,119,96,67,22,29,107,84,55),(11,73,30,56,120,23,85,42,68,108),(12,57,86,109,31,24,69,74,97,43)], [(1,19,13,7),(2,8,14,20),(3,21,15,9),(4,10,16,22),(5,23,17,11),(6,12,18,24),(25,31,37,43),(26,44,38,32),(27,33,39,45),(28,46,40,34),(29,35,41,47),(30,48,42,36),(49,55,61,67),(50,68,62,56),(51,57,63,69),(52,70,64,58),(53,59,65,71),(54,72,66,60),(73,91,85,79),(74,80,86,92),(75,93,87,81),(76,82,88,94),(77,95,89,83),(78,84,90,96),(97,103,109,115),(98,116,110,104),(99,105,111,117),(100,118,112,106),(101,107,113,119),(102,120,114,108)], [(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)], [(1,6,19,12,13,18,7,24),(2,23,8,17,14,11,20,5),(3,4,21,10,15,16,9,22),(25,38,31,32,37,26,43,44),(27,36,33,30,39,48,45,42),(28,41,46,47,40,29,34,35),(49,66,55,60,61,54,67,72),(50,71,68,53,62,59,56,65),(51,64,57,58,63,52,69,70),(73,82,91,88,85,94,79,76),(74,75,80,93,86,87,92,81),(77,78,95,84,89,90,83,96),(97,98,103,116,109,110,115,104),(99,120,105,114,111,108,117,102),(100,101,118,107,112,113,106,119)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 12A | 12B | 12C | 15 | 24A | 24B | 24C | 24D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 24 | 24 | 24 | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 12 | 60 | 2 | 10 | 10 | 4 | 2 | 4 | 20 | 20 | 60 | 60 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | 10 | 20 | 8 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | F5 | C4.D4 | C2×F5 | C22⋊F5 | C12.46D4 | C23.F5 | S3×F5 | D6⋊F5 | Dic5.D12 |
| kernel | Dic5.D12 | C3×C22.F5 | C15⋊8M4(2) | C2×C5⋊D12 | S3×C2×C10 | C22×D15 | C22.F5 | C3×Dic5 | C2×Dic5 | Dic5 | Dic5 | C2×C10 | C22×S3 | C15 | C2×C6 | C6 | C5 | C3 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of Dic5.D12 ►in GL8(𝔽241)
| 91 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 129 | 240 | 1 | 0 | 0 | 0 | 0 | 0 |
| 239 | 188 | 52 | 0 | 0 | 0 | 0 | 0 |
| 36 | 227 | 120 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 125 | 39 | 141 | 120 | 0 | 0 | 0 | 0 |
| 109 | 74 | 42 | 186 | 0 | 0 | 0 | 0 |
| 205 | 230 | 205 | 186 | 0 | 0 | 0 | 0 |
| 196 | 129 | 177 | 78 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 74 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 185 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 155 | 142 | 43 |
| 0 | 0 | 0 | 0 | 0 | 155 | 198 | 99 |
| 81 | 128 | 224 | 0 | 0 | 0 | 0 | 0 |
| 135 | 223 | 32 | 224 | 0 | 0 | 0 | 0 |
| 105 | 146 | 180 | 191 | 0 | 0 | 0 | 0 |
| 167 | 101 | 210 | 239 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 97 | 97 |
| 0 | 0 | 0 | 0 | 10 | 0 | 2 | 1 |
| 0 | 0 | 0 | 0 | 67 | 142 | 239 | 239 |
| 0 | 0 | 0 | 0 | 54 | 198 | 239 | 239 |
| 81 | 128 | 224 | 0 | 0 | 0 | 0 | 0 |
| 135 | 223 | 32 | 224 | 0 | 0 | 0 | 0 |
| 105 | 146 | 180 | 191 | 0 | 0 | 0 | 0 |
| 167 | 101 | 210 | 239 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 97 | 97 |
| 0 | 0 | 0 | 0 | 10 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 67 | 142 | 239 | 239 |
| 0 | 0 | 0 | 0 | 80 | 142 | 239 | 239 |
G:=sub<GL(8,GF(241))| [91,129,239,36,0,0,0,0,0,240,188,227,0,0,0,0,0,1,52,120,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[125,109,205,196,0,0,0,0,39,74,230,129,0,0,0,0,141,42,205,177,0,0,0,0,120,186,186,78,0,0,0,0,0,0,0,0,56,26,189,0,0,0,0,0,74,185,155,155,0,0,0,0,0,0,142,198,0,0,0,0,0,0,43,99],[81,135,105,167,0,0,0,0,128,223,146,101,0,0,0,0,224,32,180,210,0,0,0,0,0,224,191,239,0,0,0,0,0,0,0,0,4,10,67,54,0,0,0,0,0,0,142,198,0,0,0,0,97,2,239,239,0,0,0,0,97,1,239,239],[81,135,105,167,0,0,0,0,128,223,146,101,0,0,0,0,224,32,180,210,0,0,0,0,0,224,191,239,0,0,0,0,0,0,0,0,4,10,67,80,0,0,0,0,0,0,142,142,0,0,0,0,97,1,239,239,0,0,0,0,97,2,239,239] >;
Dic5.D12 in GAP, Magma, Sage, TeX
{\rm Dic}_5.D_{12} % in TeX
G:=Group("Dic5.D12"); // GroupNames label
G:=SmallGroup(480,250);
// by ID
G=gap.SmallGroup(480,250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=1,b^2=c^12=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*b*c^11>;
// generators/relations