metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.45D4, C6.89(D4×D5), D30⋊30(C2×C4), C10.90(S3×D4), (C2×Dic5)⋊13D6, C30.241(C2×D4), C23.D5⋊11S3, D30⋊4C4⋊32C2, C23.53(S3×D5), D15⋊3(C22⋊C4), (C2×Dic3)⋊13D10, C6.D4⋊11D5, (C22×D15)⋊10C4, (C6×Dic5)⋊6C22, (C23×D15).4C2, (C22×C10).55D6, (C22×C6).40D10, C2.7(D10⋊D6), (C2×C30).203C23, C30.151(C22×C4), C22⋊4(D30.C2), (C10×Dic3)⋊6C22, (C22×C30).65C22, (C22×D15).113C22, (C2×C6)⋊4(C4×D5), C5⋊3(S3×C22⋊C4), C3⋊2(D5×C22⋊C4), C6.56(C2×C4×D5), C10.88(S3×C2×C4), (C2×C10)⋊13(C4×S3), (C2×C30)⋊22(C2×C4), C15⋊13(C2×C22⋊C4), C22.91(C2×S3×D5), (C2×D30.C2)⋊17C2, C2.20(C2×D30.C2), (C3×C23.D5)⋊13C2, (C5×C6.D4)⋊13C2, (C2×C6).215(C22×D5), (C2×C10).215(C22×S3), SmallGroup(480,637)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.45D4
G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=dbd-1=a18b, dcd-1=a15c-1 >
Subgroups: 1740 in 264 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, D15, D15, C30, C30, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C10, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C5×Dic3, C3×Dic5, D30, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, S3×C22⋊C4, D30.C2, C6×Dic5, C10×Dic3, C22×D15, C22×D15, C22×D15, C22×C30, D5×C22⋊C4, D30⋊4C4, C3×C23.D5, C5×C6.D4, C2×D30.C2, C23×D15, D30.45D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, D10, C4×S3, C22×S3, C2×C22⋊C4, C4×D5, C22×D5, S3×C2×C4, S3×D4, S3×D5, C2×C4×D5, D4×D5, S3×C22⋊C4, D30.C2, C2×S3×D5, D5×C22⋊C4, C2×D30.C2, D10⋊D6, D30.45D4
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 90)(82 89)(83 88)(84 87)(85 86)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 97 71 54)(2 116 72 43)(3 105 73 32)(4 94 74 51)(5 113 75 40)(6 102 76 59)(7 91 77 48)(8 110 78 37)(9 99 79 56)(10 118 80 45)(11 107 81 34)(12 96 82 53)(13 115 83 42)(14 104 84 31)(15 93 85 50)(16 112 86 39)(17 101 87 58)(18 120 88 47)(19 109 89 36)(20 98 90 55)(21 117 61 44)(22 106 62 33)(23 95 63 52)(24 114 64 41)(25 103 65 60)(26 92 66 49)(27 111 67 38)(28 100 68 57)(29 119 69 46)(30 108 70 35)
(1 112 16 97)(2 101 17 116)(3 120 18 105)(4 109 19 94)(5 98 20 113)(6 117 21 102)(7 106 22 91)(8 95 23 110)(9 114 24 99)(10 103 25 118)(11 92 26 107)(12 111 27 96)(13 100 28 115)(14 119 29 104)(15 108 30 93)(31 84 46 69)(32 73 47 88)(33 62 48 77)(34 81 49 66)(35 70 50 85)(36 89 51 74)(37 78 52 63)(38 67 53 82)(39 86 54 71)(40 75 55 90)(41 64 56 79)(42 83 57 68)(43 72 58 87)(44 61 59 76)(45 80 60 65)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,97,71,54)(2,116,72,43)(3,105,73,32)(4,94,74,51)(5,113,75,40)(6,102,76,59)(7,91,77,48)(8,110,78,37)(9,99,79,56)(10,118,80,45)(11,107,81,34)(12,96,82,53)(13,115,83,42)(14,104,84,31)(15,93,85,50)(16,112,86,39)(17,101,87,58)(18,120,88,47)(19,109,89,36)(20,98,90,55)(21,117,61,44)(22,106,62,33)(23,95,63,52)(24,114,64,41)(25,103,65,60)(26,92,66,49)(27,111,67,38)(28,100,68,57)(29,119,69,46)(30,108,70,35), (1,112,16,97)(2,101,17,116)(3,120,18,105)(4,109,19,94)(5,98,20,113)(6,117,21,102)(7,106,22,91)(8,95,23,110)(9,114,24,99)(10,103,25,118)(11,92,26,107)(12,111,27,96)(13,100,28,115)(14,119,29,104)(15,108,30,93)(31,84,46,69)(32,73,47,88)(33,62,48,77)(34,81,49,66)(35,70,50,85)(36,89,51,74)(37,78,52,63)(38,67,53,82)(39,86,54,71)(40,75,55,90)(41,64,56,79)(42,83,57,68)(43,72,58,87)(44,61,59,76)(45,80,60,65)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,97,71,54)(2,116,72,43)(3,105,73,32)(4,94,74,51)(5,113,75,40)(6,102,76,59)(7,91,77,48)(8,110,78,37)(9,99,79,56)(10,118,80,45)(11,107,81,34)(12,96,82,53)(13,115,83,42)(14,104,84,31)(15,93,85,50)(16,112,86,39)(17,101,87,58)(18,120,88,47)(19,109,89,36)(20,98,90,55)(21,117,61,44)(22,106,62,33)(23,95,63,52)(24,114,64,41)(25,103,65,60)(26,92,66,49)(27,111,67,38)(28,100,68,57)(29,119,69,46)(30,108,70,35), (1,112,16,97)(2,101,17,116)(3,120,18,105)(4,109,19,94)(5,98,20,113)(6,117,21,102)(7,106,22,91)(8,95,23,110)(9,114,24,99)(10,103,25,118)(11,92,26,107)(12,111,27,96)(13,100,28,115)(14,119,29,104)(15,108,30,93)(31,84,46,69)(32,73,47,88)(33,62,48,77)(34,81,49,66)(35,70,50,85)(36,89,51,74)(37,78,52,63)(38,67,53,82)(39,86,54,71)(40,75,55,90)(41,64,56,79)(42,83,57,68)(43,72,58,87)(44,61,59,76)(45,80,60,65) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,90),(82,89),(83,88),(84,87),(85,86),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,97,71,54),(2,116,72,43),(3,105,73,32),(4,94,74,51),(5,113,75,40),(6,102,76,59),(7,91,77,48),(8,110,78,37),(9,99,79,56),(10,118,80,45),(11,107,81,34),(12,96,82,53),(13,115,83,42),(14,104,84,31),(15,93,85,50),(16,112,86,39),(17,101,87,58),(18,120,88,47),(19,109,89,36),(20,98,90,55),(21,117,61,44),(22,106,62,33),(23,95,63,52),(24,114,64,41),(25,103,65,60),(26,92,66,49),(27,111,67,38),(28,100,68,57),(29,119,69,46),(30,108,70,35)], [(1,112,16,97),(2,101,17,116),(3,120,18,105),(4,109,19,94),(5,98,20,113),(6,117,21,102),(7,106,22,91),(8,95,23,110),(9,114,24,99),(10,103,25,118),(11,92,26,107),(12,111,27,96),(13,100,28,115),(14,119,29,104),(15,108,30,93),(31,84,46,69),(32,73,47,88),(33,62,48,77),(34,81,49,66),(35,70,50,85),(36,89,51,74),(37,78,52,63),(38,67,53,82),(39,86,54,71),(40,75,55,90),(41,64,56,79),(42,83,57,68),(43,72,58,87),(44,61,59,76),(45,80,60,65)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | C4×S3 | C4×D5 | S3×D4 | S3×D5 | D4×D5 | D30.C2 | C2×S3×D5 | D10⋊D6 |
| kernel | D30.45D4 | D30⋊4C4 | C3×C23.D5 | C5×C6.D4 | C2×D30.C2 | C23×D15 | C22×D15 | C23.D5 | D30 | C6.D4 | C2×Dic5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 4 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of D30.45D4 ►in GL6(𝔽61)
| 43 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 56 | 0 | 0 |
| 0 | 0 | 25 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 56 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 52 |
| 0 | 0 | 0 | 0 | 54 | 11 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 52 |
| 0 | 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(6,GF(61))| [43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,25,0,0,0,0,56,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[43,18,0,0,0,0,60,18,0,0,0,0,0,0,60,0,0,0,0,0,56,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,54,0,0,0,0,52,11],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,52,11] >;
D30.45D4 in GAP, Magma, Sage, TeX
D_{30}._{45}D_4 % in TeX
G:=Group("D30.45D4"); // GroupNames label
G:=SmallGroup(480,637);
// by ID
G=gap.SmallGroup(480,637);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=d*b*d^-1=a^18*b,d*c*d^-1=a^15*c^-1>;
// generators/relations