metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊2Dic3, Q8⋊2(C3⋊F5), (C3×Q8)⋊1F5, (Q8×C15)⋊1C4, C5⋊(Q8⋊2Dic3), (Q8×D5).4S3, C60.35(C2×C4), (C5×Q8)⋊4Dic3, C3⋊3(Q8⋊F5), (C4×D5).29D6, (C6×D5).77D4, C12.11(C2×F5), (C3×D5).6Q16, C60⋊C4.5C2, (C3×Dic10)⋊2C4, C15⋊9(Q8⋊C4), C20.3(C2×Dic3), C60.C4.4C2, (C3×Dic5).38D4, (C3×D5).10SD16, D5.3(C3⋊Q16), C6.22(C22⋊F5), D10.34(C3⋊D4), C30.22(C22⋊C4), D5.3(Q8⋊2S3), Dic5.7(C3⋊D4), (D5×C12).70C22, C10.7(C6.D4), C2.8(D10.D6), C4.3(C2×C3⋊F5), (C3×Q8×D5).2C2, SmallGroup(480,314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊2Dic3
G = < a,b,c,d | a20=c6=1, b2=a10, d2=c3, bab-1=a-1, cac-1=a9, dad-1=a7, bc=cb, dbd-1=a15b, dcd-1=c-1 >
Subgroups: 444 in 84 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, Q8, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C3⋊C8, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, Q8⋊C4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C3⋊F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, Q8⋊2Dic3, C15⋊C8, C3×Dic10, C3×Dic10, D5×C12, D5×C12, Q8×C15, C2×C3⋊F5, Q8⋊F5, C60.C4, C60⋊C4, C3×Q8×D5, Dic10⋊2Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, F5, C2×Dic3, C3⋊D4, Q8⋊C4, C2×F5, Q8⋊2S3, C3⋊Q16, C6.D4, C3⋊F5, C22⋊F5, Q8⋊2Dic3, C2×C3⋊F5, Q8⋊F5, D10.D6, Dic10⋊2Dic3
►On 120 points
Generators in S120
(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 44 11 54)(2 43 12 53)(3 42 13 52)(4 41 14 51)(5 60 15 50)(6 59 16 49)(7 58 17 48)(8 57 18 47)(9 56 19 46)(10 55 20 45)(21 84 31 94)(22 83 32 93)(23 82 33 92)(24 81 34 91)(25 100 35 90)(26 99 36 89)(27 98 37 88)(28 97 38 87)(29 96 39 86)(30 95 40 85)(61 104 71 114)(62 103 72 113)(63 102 73 112)(64 101 74 111)(65 120 75 110)(66 119 76 109)(67 118 77 108)(68 117 78 107)(69 116 79 106)(70 115 80 105)
(1 103 96)(2 112 97 10 104 85)(3 101 98 19 105 94)(4 110 99 8 106 83)(5 119 100 17 107 92)(6 108 81)(7 117 82 15 109 90)(9 115 84 13 111 88)(11 113 86)(12 102 87 20 114 95)(14 120 89 18 116 93)(16 118 91)(21 42 74 37 46 70)(22 51 75 26 47 79)(23 60 76 35 48 68)(24 49 77)(25 58 78 33 50 66)(27 56 80 31 52 64)(28 45 61 40 53 73)(29 54 62)(30 43 63 38 55 71)(32 41 65 36 57 69)(34 59 67)(39 44 72)
(2 4 10 8)(3 7 19 15)(5 13 17 9)(6 16)(12 14 20 18)(21 63 37 71)(22 66 26 78)(23 69 35 65)(24 72)(25 75 33 79)(27 61 31 73)(28 64 40 80)(29 67)(30 70 38 74)(32 76 36 68)(34 62)(39 77)(41 60 57 48)(42 43 46 55)(44 49)(45 52 53 56)(47 58 51 50)(54 59)(81 118)(82 101 90 105)(83 104 99 112)(84 107 88 119)(85 110 97 106)(86 113)(87 116 95 120)(89 102 93 114)(91 108)(92 111 100 115)(94 117 98 109)(96 103)
G:=sub<Sym(120)| (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), (1,44,11,54)(2,43,12,53)(3,42,13,52)(4,41,14,51)(5,60,15,50)(6,59,16,49)(7,58,17,48)(8,57,18,47)(9,56,19,46)(10,55,20,45)(21,84,31,94)(22,83,32,93)(23,82,33,92)(24,81,34,91)(25,100,35,90)(26,99,36,89)(27,98,37,88)(28,97,38,87)(29,96,39,86)(30,95,40,85)(61,104,71,114)(62,103,72,113)(63,102,73,112)(64,101,74,111)(65,120,75,110)(66,119,76,109)(67,118,77,108)(68,117,78,107)(69,116,79,106)(70,115,80,105), (1,103,96)(2,112,97,10,104,85)(3,101,98,19,105,94)(4,110,99,8,106,83)(5,119,100,17,107,92)(6,108,81)(7,117,82,15,109,90)(9,115,84,13,111,88)(11,113,86)(12,102,87,20,114,95)(14,120,89,18,116,93)(16,118,91)(21,42,74,37,46,70)(22,51,75,26,47,79)(23,60,76,35,48,68)(24,49,77)(25,58,78,33,50,66)(27,56,80,31,52,64)(28,45,61,40,53,73)(29,54,62)(30,43,63,38,55,71)(32,41,65,36,57,69)(34,59,67)(39,44,72), (2,4,10,8)(3,7,19,15)(5,13,17,9)(6,16)(12,14,20,18)(21,63,37,71)(22,66,26,78)(23,69,35,65)(24,72)(25,75,33,79)(27,61,31,73)(28,64,40,80)(29,67)(30,70,38,74)(32,76,36,68)(34,62)(39,77)(41,60,57,48)(42,43,46,55)(44,49)(45,52,53,56)(47,58,51,50)(54,59)(81,118)(82,101,90,105)(83,104,99,112)(84,107,88,119)(85,110,97,106)(86,113)(87,116,95,120)(89,102,93,114)(91,108)(92,111,100,115)(94,117,98,109)(96,103)>;
G:=Group( (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), (1,44,11,54)(2,43,12,53)(3,42,13,52)(4,41,14,51)(5,60,15,50)(6,59,16,49)(7,58,17,48)(8,57,18,47)(9,56,19,46)(10,55,20,45)(21,84,31,94)(22,83,32,93)(23,82,33,92)(24,81,34,91)(25,100,35,90)(26,99,36,89)(27,98,37,88)(28,97,38,87)(29,96,39,86)(30,95,40,85)(61,104,71,114)(62,103,72,113)(63,102,73,112)(64,101,74,111)(65,120,75,110)(66,119,76,109)(67,118,77,108)(68,117,78,107)(69,116,79,106)(70,115,80,105), (1,103,96)(2,112,97,10,104,85)(3,101,98,19,105,94)(4,110,99,8,106,83)(5,119,100,17,107,92)(6,108,81)(7,117,82,15,109,90)(9,115,84,13,111,88)(11,113,86)(12,102,87,20,114,95)(14,120,89,18,116,93)(16,118,91)(21,42,74,37,46,70)(22,51,75,26,47,79)(23,60,76,35,48,68)(24,49,77)(25,58,78,33,50,66)(27,56,80,31,52,64)(28,45,61,40,53,73)(29,54,62)(30,43,63,38,55,71)(32,41,65,36,57,69)(34,59,67)(39,44,72), (2,4,10,8)(3,7,19,15)(5,13,17,9)(6,16)(12,14,20,18)(21,63,37,71)(22,66,26,78)(23,69,35,65)(24,72)(25,75,33,79)(27,61,31,73)(28,64,40,80)(29,67)(30,70,38,74)(32,76,36,68)(34,62)(39,77)(41,60,57,48)(42,43,46,55)(44,49)(45,52,53,56)(47,58,51,50)(54,59)(81,118)(82,101,90,105)(83,104,99,112)(84,107,88,119)(85,110,97,106)(86,113)(87,116,95,120)(89,102,93,114)(91,108)(92,111,100,115)(94,117,98,109)(96,103) );
G=PermutationGroup([[(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)], [(1,44,11,54),(2,43,12,53),(3,42,13,52),(4,41,14,51),(5,60,15,50),(6,59,16,49),(7,58,17,48),(8,57,18,47),(9,56,19,46),(10,55,20,45),(21,84,31,94),(22,83,32,93),(23,82,33,92),(24,81,34,91),(25,100,35,90),(26,99,36,89),(27,98,37,88),(28,97,38,87),(29,96,39,86),(30,95,40,85),(61,104,71,114),(62,103,72,113),(63,102,73,112),(64,101,74,111),(65,120,75,110),(66,119,76,109),(67,118,77,108),(68,117,78,107),(69,116,79,106),(70,115,80,105)], [(1,103,96),(2,112,97,10,104,85),(3,101,98,19,105,94),(4,110,99,8,106,83),(5,119,100,17,107,92),(6,108,81),(7,117,82,15,109,90),(9,115,84,13,111,88),(11,113,86),(12,102,87,20,114,95),(14,120,89,18,116,93),(16,118,91),(21,42,74,37,46,70),(22,51,75,26,47,79),(23,60,76,35,48,68),(24,49,77),(25,58,78,33,50,66),(27,56,80,31,52,64),(28,45,61,40,53,73),(29,54,62),(30,43,63,38,55,71),(32,41,65,36,57,69),(34,59,67),(39,44,72)], [(2,4,10,8),(3,7,19,15),(5,13,17,9),(6,16),(12,14,20,18),(21,63,37,71),(22,66,26,78),(23,69,35,65),(24,72),(25,75,33,79),(27,61,31,73),(28,64,40,80),(29,67),(30,70,38,74),(32,76,36,68),(34,62),(39,77),(41,60,57,48),(42,43,46,55),(44,49),(45,52,53,56),(47,58,51,50),(54,59),(81,118),(82,101,90,105),(83,104,99,112),(84,107,88,119),(85,110,97,106),(86,113),(87,116,95,120),(89,102,93,114),(91,108),(92,111,100,115),(94,117,98,109),(96,103)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 30A | 30B | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 4 | 10 | 20 | 60 | 60 | 4 | 2 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | ··· | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | - | - | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | Dic3 | D6 | Dic3 | SD16 | Q16 | C3⋊D4 | C3⋊D4 | F5 | C2×F5 | Q8⋊2S3 | C3⋊Q16 | C3⋊F5 | C22⋊F5 | C2×C3⋊F5 | D10.D6 | Q8⋊F5 | Dic10⋊2Dic3 |
| kernel | Dic10⋊2Dic3 | C60.C4 | C60⋊C4 | C3×Q8×D5 | C3×Dic10 | Q8×C15 | Q8×D5 | C3×Dic5 | C6×D5 | Dic10 | C4×D5 | C5×Q8 | C3×D5 | C3×D5 | Dic5 | D10 | C3×Q8 | C12 | D5 | D5 | Q8 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of Dic10⋊2Dic3 ►in GL6(𝔽241)
| 240 | 3 | 0 | 0 | 0 | 0 |
| 160 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 184 | 0 | 0 | 0 | 0 |
| 148 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 126 |
| 0 | 0 | 0 | 229 | 114 | 229 |
| 0 | 0 | 229 | 114 | 229 | 0 |
| 0 | 0 | 126 | 0 | 12 | 12 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 118 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(241))| [240,160,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,148,0,0,0,0,184,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,12,0,229,126,0,0,12,229,114,0,0,0,0,114,229,12,0,0,126,229,0,12],[177,118,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0] >;
Dic10⋊2Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_2{\rm Dic}_3 % in TeX
G:=Group("Dic10:2Dic3"); // GroupNames label
G:=SmallGroup(480,314);
// by ID
G=gap.SmallGroup(480,314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^6=1,b^2=a^10,d^2=c^3,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations