metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.8D4, D20.6D6, D12.6D10, C60.6C23, Dic15.40D4, Dic30⋊1C22, C3⋊C8⋊6D10, D4⋊D5⋊3S3, D4⋊S3⋊3D5, D4⋊6(S3×D5), (C5×D4)⋊4D6, C5⋊2C8⋊6D6, (C3×D4)⋊4D10, C5⋊3(D8⋊S3), C6.68(D4×D5), C20⋊D6⋊2C2, C3⋊3(D8⋊D5), C10.69(S3×D4), D4⋊2D15⋊1C2, C15⋊15(C8⋊C22), D12.D5⋊1C2, C6.D20⋊1C2, C30.168(C2×D4), (D4×C15)⋊6C22, C20.6(C22×S3), C12.6(C22×D5), D30.5C4⋊1C2, (C5×D12).2C22, (C4×D15).2C22, (C3×D20).2C22, C2.21(D10⋊D6), C4.6(C2×S3×D5), (C3×D4⋊D5)⋊4C2, (C5×D4⋊S3)⋊4C2, (C5×C3⋊C8)⋊4C22, (C3×C5⋊2C8)⋊4C22, SmallGroup(480,558)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.8D4
G = < a,b,c,d | a20=b2=c6=1, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a15b, dcd-1=a5c-1 >
Subgroups: 940 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, Dic15, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, C15⋊D4, C3×D20, C5×D12, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, C2×S3×D5, D8⋊D5, D30.5C4, C6.D20, D12.D5, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D4⋊2D15, D30.8D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D8⋊D5, D10⋊D6, D30.8D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 100)(95 99)(96 98)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)
(1 95 73 28 47 117)(2 86 74 39 48 108)(3 97 75 30 49 119)(4 88 76 21 50 110)(5 99 77 32 51 101)(6 90 78 23 52 112)(7 81 79 34 53 103)(8 92 80 25 54 114)(9 83 61 36 55 105)(10 94 62 27 56 116)(11 85 63 38 57 107)(12 96 64 29 58 118)(13 87 65 40 59 109)(14 98 66 31 60 120)(15 89 67 22 41 111)(16 100 68 33 42 102)(17 91 69 24 43 113)(18 82 70 35 44 104)(19 93 71 26 45 115)(20 84 72 37 46 106)
(1 117 6 102 11 107 16 112)(2 118 7 103 12 108 17 113)(3 119 8 104 13 109 18 114)(4 120 9 105 14 110 19 115)(5 101 10 106 15 111 20 116)(21 71 26 76 31 61 36 66)(22 72 27 77 32 62 37 67)(23 73 28 78 33 63 38 68)(24 74 29 79 34 64 39 69)(25 75 30 80 35 65 40 70)(41 89 46 94 51 99 56 84)(42 90 47 95 52 100 57 85)(43 91 48 96 53 81 58 86)(44 92 49 97 54 82 59 87)(45 93 50 98 55 83 60 88)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120), (1,95,73,28,47,117)(2,86,74,39,48,108)(3,97,75,30,49,119)(4,88,76,21,50,110)(5,99,77,32,51,101)(6,90,78,23,52,112)(7,81,79,34,53,103)(8,92,80,25,54,114)(9,83,61,36,55,105)(10,94,62,27,56,116)(11,85,63,38,57,107)(12,96,64,29,58,118)(13,87,65,40,59,109)(14,98,66,31,60,120)(15,89,67,22,41,111)(16,100,68,33,42,102)(17,91,69,24,43,113)(18,82,70,35,44,104)(19,93,71,26,45,115)(20,84,72,37,46,106), (1,117,6,102,11,107,16,112)(2,118,7,103,12,108,17,113)(3,119,8,104,13,109,18,114)(4,120,9,105,14,110,19,115)(5,101,10,106,15,111,20,116)(21,71,26,76,31,61,36,66)(22,72,27,77,32,62,37,67)(23,73,28,78,33,63,38,68)(24,74,29,79,34,64,39,69)(25,75,30,80,35,65,40,70)(41,89,46,94,51,99,56,84)(42,90,47,95,52,100,57,85)(43,91,48,96,53,81,58,86)(44,92,49,97,54,82,59,87)(45,93,50,98,55,83,60,88)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,64)(62,63)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,100)(95,99)(96,98)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120), (1,95,73,28,47,117)(2,86,74,39,48,108)(3,97,75,30,49,119)(4,88,76,21,50,110)(5,99,77,32,51,101)(6,90,78,23,52,112)(7,81,79,34,53,103)(8,92,80,25,54,114)(9,83,61,36,55,105)(10,94,62,27,56,116)(11,85,63,38,57,107)(12,96,64,29,58,118)(13,87,65,40,59,109)(14,98,66,31,60,120)(15,89,67,22,41,111)(16,100,68,33,42,102)(17,91,69,24,43,113)(18,82,70,35,44,104)(19,93,71,26,45,115)(20,84,72,37,46,106), (1,117,6,102,11,107,16,112)(2,118,7,103,12,108,17,113)(3,119,8,104,13,109,18,114)(4,120,9,105,14,110,19,115)(5,101,10,106,15,111,20,116)(21,71,26,76,31,61,36,66)(22,72,27,77,32,62,37,67)(23,73,28,78,33,63,38,68)(24,74,29,79,34,64,39,69)(25,75,30,80,35,65,40,70)(41,89,46,94,51,99,56,84)(42,90,47,95,52,100,57,85)(43,91,48,96,53,81,58,86)(44,92,49,97,54,82,59,87)(45,93,50,98,55,83,60,88) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,64),(62,63),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,100),(95,99),(96,98),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120)], [(1,95,73,28,47,117),(2,86,74,39,48,108),(3,97,75,30,49,119),(4,88,76,21,50,110),(5,99,77,32,51,101),(6,90,78,23,52,112),(7,81,79,34,53,103),(8,92,80,25,54,114),(9,83,61,36,55,105),(10,94,62,27,56,116),(11,85,63,38,57,107),(12,96,64,29,58,118),(13,87,65,40,59,109),(14,98,66,31,60,120),(15,89,67,22,41,111),(16,100,68,33,42,102),(17,91,69,24,43,113),(18,82,70,35,44,104),(19,93,71,26,45,115),(20,84,72,37,46,106)], [(1,117,6,102,11,107,16,112),(2,118,7,103,12,108,17,113),(3,119,8,104,13,109,18,114),(4,120,9,105,14,110,19,115),(5,101,10,106,15,111,20,116),(21,71,26,76,31,61,36,66),(22,72,27,77,32,62,37,67),(23,73,28,78,33,63,38,68),(24,74,29,79,34,64,39,69),(25,75,30,80,35,65,40,70),(41,89,46,94,51,99,56,84),(42,90,47,95,52,100,57,85),(43,91,48,96,53,81,58,86),(44,92,49,97,54,82,59,87),(45,93,50,98,55,83,60,88)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 12 | 20 | 30 | 2 | 2 | 30 | 60 | 2 | 2 | 2 | 8 | 40 | 12 | 20 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | D8⋊S3 | C2×S3×D5 | D8⋊D5 | D10⋊D6 | D30.8D4 |
| kernel | D30.8D4 | D30.5C4 | C6.D20 | D12.D5 | C3×D4⋊D5 | C5×D4⋊S3 | C20⋊D6 | D4⋊2D15 | D4⋊D5 | Dic15 | D30 | D4⋊S3 | C5⋊2C8 | D20 | C5×D4 | C3⋊C8 | D12 | C3×D4 | C15 | C10 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D30.8D4 ►in GL6(𝔽241)
| 189 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 189 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 194 | 47 | 194 | 47 |
| 0 | 0 | 194 | 147 | 194 | 147 |
| 0 | 0 | 194 | 47 | 47 | 194 |
| 0 | 0 | 194 | 147 | 47 | 94 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 194 | 194 | 47 |
| 0 | 0 | 147 | 194 | 94 | 47 |
| 0 | 0 | 47 | 194 | 47 | 194 |
| 0 | 0 | 147 | 194 | 147 | 194 |
G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,189,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,194,194,194,194,0,0,47,147,47,147,0,0,194,194,47,47,0,0,47,147,194,94],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,194,194,194,194,0,0,194,94,47,147,0,0,47,47,194,194] >;
D30.8D4 in GAP, Magma, Sage, TeX
D_{30}._8D_4 % in TeX
G:=Group("D30.8D4"); // GroupNames label
G:=SmallGroup(480,558);
// by ID
G=gap.SmallGroup(480,558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations