metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊8D30, Q8⋊8D30, D60⋊28C22, C60.89C23, C30.65C24, D30.30C23, C15⋊122+ 1+4, Dic30⋊39C22, Dic15.32C23, (C2×C4)⋊4D30, C4○D4⋊5D15, (C5×D4)⋊24D6, (C2×C20)⋊14D6, C5⋊5(D4○D12), (C5×Q8)⋊24D6, (C3×D4)⋊24D10, (C2×D60)⋊16C2, (D4×D15)⋊12C2, (C2×C12)⋊14D10, (C3×Q8)⋊21D10, C3⋊5(D4⋊8D10), (C2×C60)⋊10C22, Q8⋊3D15⋊12C2, C6.65(C23×D5), (C4×D15)⋊11C22, (D4×C15)⋊26C22, C15⋊7D4⋊12C22, C10.65(S3×C23), (C2×C30).11C23, D60⋊11C2⋊18C2, (Q8×C15)⋊23C22, C4.32(C22×D15), C2.13(C23×D15), C20.139(C22×S3), C12.137(C22×D5), (C22×D15)⋊4C22, C22.3(C22×D15), (C3×C4○D4)⋊4D5, (C5×C4○D4)⋊8S3, (C15×C4○D4)⋊4C2, (C2×C6).18(C22×D5), (C2×C10).19(C22×S3), SmallGroup(480,1176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊8D30
G = < a,b,c,d | a4=b2=c30=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >
Subgroups: 2180 in 332 conjugacy classes, 119 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, Dic15, C60, C60, D30, D30, C2×C30, C2×D20, C4○D20, D4×D5, Q8⋊2D5, C5×C4○D4, D4○D12, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D4×C15, Q8×C15, C22×D15, D4⋊8D10, C2×D60, D60⋊11C2, D4×D15, Q8⋊3D15, C15×C4○D4, D4⋊8D30
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, D15, 2+ 1+4, C22×D5, S3×C23, D30, C23×D5, D4○D12, C22×D15, D4⋊8D10, C23×D15, D4⋊8D30
►On 120 points
Generators in S120
(1 116 72 36)(2 117 73 37)(3 118 74 38)(4 119 75 39)(5 120 76 40)(6 91 77 41)(7 92 78 42)(8 93 79 43)(9 94 80 44)(10 95 81 45)(11 96 82 46)(12 97 83 47)(13 98 84 48)(14 99 85 49)(15 100 86 50)(16 101 87 51)(17 102 88 52)(18 103 89 53)(19 104 90 54)(20 105 61 55)(21 106 62 56)(22 107 63 57)(23 108 64 58)(24 109 65 59)(25 110 66 60)(26 111 67 31)(27 112 68 32)(28 113 69 33)(29 114 70 34)(30 115 71 35)
(1 36)(2 117)(3 38)(4 119)(5 40)(6 91)(7 42)(8 93)(9 44)(10 95)(11 46)(12 97)(13 48)(14 99)(15 50)(16 101)(17 52)(18 103)(19 54)(20 105)(21 56)(22 107)(23 58)(24 109)(25 60)(26 111)(27 32)(28 113)(29 34)(30 115)(31 67)(33 69)(35 71)(37 73)(39 75)(41 77)(43 79)(45 81)(47 83)(49 85)(51 87)(53 89)(55 61)(57 63)(59 65)(62 106)(64 108)(66 110)(68 112)(70 114)(72 116)(74 118)(76 120)(78 92)(80 94)(82 96)(84 98)(86 100)(88 102)(90 104)
(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)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(83 90)(84 89)(85 88)(86 87)
G:=sub<Sym(120)| (1,116,72,36)(2,117,73,37)(3,118,74,38)(4,119,75,39)(5,120,76,40)(6,91,77,41)(7,92,78,42)(8,93,79,43)(9,94,80,44)(10,95,81,45)(11,96,82,46)(12,97,83,47)(13,98,84,48)(14,99,85,49)(15,100,86,50)(16,101,87,51)(17,102,88,52)(18,103,89,53)(19,104,90,54)(20,105,61,55)(21,106,62,56)(22,107,63,57)(23,108,64,58)(24,109,65,59)(25,110,66,60)(26,111,67,31)(27,112,68,32)(28,113,69,33)(29,114,70,34)(30,115,71,35), (1,36)(2,117)(3,38)(4,119)(5,40)(6,91)(7,42)(8,93)(9,44)(10,95)(11,46)(12,97)(13,48)(14,99)(15,50)(16,101)(17,52)(18,103)(19,54)(20,105)(21,56)(22,107)(23,58)(24,109)(25,60)(26,111)(27,32)(28,113)(29,34)(30,115)(31,67)(33,69)(35,71)(37,73)(39,75)(41,77)(43,79)(45,81)(47,83)(49,85)(51,87)(53,89)(55,61)(57,63)(59,65)(62,106)(64,108)(66,110)(68,112)(70,114)(72,116)(74,118)(76,120)(78,92)(80,94)(82,96)(84,98)(86,100)(88,102)(90,104), (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)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(83,90)(84,89)(85,88)(86,87)>;
G:=Group( (1,116,72,36)(2,117,73,37)(3,118,74,38)(4,119,75,39)(5,120,76,40)(6,91,77,41)(7,92,78,42)(8,93,79,43)(9,94,80,44)(10,95,81,45)(11,96,82,46)(12,97,83,47)(13,98,84,48)(14,99,85,49)(15,100,86,50)(16,101,87,51)(17,102,88,52)(18,103,89,53)(19,104,90,54)(20,105,61,55)(21,106,62,56)(22,107,63,57)(23,108,64,58)(24,109,65,59)(25,110,66,60)(26,111,67,31)(27,112,68,32)(28,113,69,33)(29,114,70,34)(30,115,71,35), (1,36)(2,117)(3,38)(4,119)(5,40)(6,91)(7,42)(8,93)(9,44)(10,95)(11,46)(12,97)(13,48)(14,99)(15,50)(16,101)(17,52)(18,103)(19,54)(20,105)(21,56)(22,107)(23,58)(24,109)(25,60)(26,111)(27,32)(28,113)(29,34)(30,115)(31,67)(33,69)(35,71)(37,73)(39,75)(41,77)(43,79)(45,81)(47,83)(49,85)(51,87)(53,89)(55,61)(57,63)(59,65)(62,106)(64,108)(66,110)(68,112)(70,114)(72,116)(74,118)(76,120)(78,92)(80,94)(82,96)(84,98)(86,100)(88,102)(90,104), (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)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(83,90)(84,89)(85,88)(86,87) );
G=PermutationGroup([[(1,116,72,36),(2,117,73,37),(3,118,74,38),(4,119,75,39),(5,120,76,40),(6,91,77,41),(7,92,78,42),(8,93,79,43),(9,94,80,44),(10,95,81,45),(11,96,82,46),(12,97,83,47),(13,98,84,48),(14,99,85,49),(15,100,86,50),(16,101,87,51),(17,102,88,52),(18,103,89,53),(19,104,90,54),(20,105,61,55),(21,106,62,56),(22,107,63,57),(23,108,64,58),(24,109,65,59),(25,110,66,60),(26,111,67,31),(27,112,68,32),(28,113,69,33),(29,114,70,34),(30,115,71,35)], [(1,36),(2,117),(3,38),(4,119),(5,40),(6,91),(7,42),(8,93),(9,44),(10,95),(11,46),(12,97),(13,48),(14,99),(15,50),(16,101),(17,52),(18,103),(19,54),(20,105),(21,56),(22,107),(23,58),(24,109),(25,60),(26,111),(27,32),(28,113),(29,34),(30,115),(31,67),(33,69),(35,71),(37,73),(39,75),(41,77),(43,79),(45,81),(47,83),(49,85),(51,87),(53,89),(55,61),(57,63),(59,65),(62,106),(64,108),(66,110),(68,112),(70,114),(72,116),(74,118),(76,120),(78,92),(80,94),(82,96),(84,98),(86,100),(88,102),(90,104)], [(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),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,114),(38,113),(39,112),(40,111),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,100),(52,99),(53,98),(54,97),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(83,90),(84,89),(85,88),(86,87)]])
87 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | 2+ 1+4 | D4○D12 | D4⋊8D10 | D4⋊8D30 |
| kernel | D4⋊8D30 | C2×D60 | D60⋊11C2 | D4×D15 | Q8⋊3D15 | C15×C4○D4 | C5×C4○D4 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 2 | 3 | 3 | 1 | 6 | 6 | 2 | 4 | 12 | 12 | 4 | 1 | 2 | 4 | 8 |
Matrix representation of D4⋊8D30 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 7 |
| 0 | 0 | 0 | 0 | 54 | 29 |
| 0 | 0 | 32 | 7 | 0 | 0 |
| 0 | 0 | 54 | 29 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 7 |
| 0 | 0 | 0 | 0 | 54 | 29 |
| 0 | 0 | 29 | 54 | 0 | 0 |
| 0 | 0 | 7 | 32 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 60 |
| 0 | 0 | 0 | 0 | 44 | 44 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,32,54,0,0,0,0,7,29,0,0,32,54,0,0,0,0,7,29,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,29,7,0,0,0,0,54,32,0,0,32,54,0,0,0,0,7,29,0,0],[1,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,60,0,0,0,17,1,0,0,0,0,60,0,0,0],[1,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,44,0,0,0,0,60,44,0,0,17,44,0,0,0,0,60,44,0,0] >;
D4⋊8D30 in GAP, Magma, Sage, TeX
D_4\rtimes_8D_{30} % in TeX
G:=Group("D4:8D30"); // GroupNames label
G:=SmallGroup(480,1176);
// by ID
G=gap.SmallGroup(480,1176);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^30=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations