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, (D4×D15)⋊12C2, (C2×D60)⋊16C2, (C3×D4)⋊24D10, (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, C2.13(C23×D15), C4.32(C22×D15), C20.139(C22×S3), C12.137(C22×D5), (C22×D15)⋊4C22, C22.3(C22×D15), (C5×C4○D4)⋊8S3, (C3×C4○D4)⋊4D5, (C15×C4○D4)⋊4C2, (C2×C6).18(C22×D5), (C2×C10).19(C22×S3), SmallGroup(480,1176)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2180 in 332 conjugacy classes, 119 normal (22 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], D5 [×6], C10, C10 [×3], Dic3 [×2], C12, C12 [×3], D6 [×12], C2×C6 [×3], C15, C2×D4 [×9], C4○D4, C4○D4 [×5], Dic5 [×2], C20, C20 [×3], D10 [×12], C2×C10 [×3], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], D15 [×6], C30, C30 [×3], 2+ (1+4), Dic10, C4×D5 [×6], D20 [×9], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q8⋊3S3 [×2], C3×C4○D4, Dic15 [×2], C60, C60 [×3], D30 [×6], D30 [×6], C2×C30 [×3], C2×D20 [×3], C4○D20 [×3], D4×D5 [×6], Q8⋊2D5 [×2], C5×C4○D4, D4○D12, Dic30, C4×D15 [×6], D60 [×9], C15⋊7D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, C22×D15 [×6], D4⋊8D10, C2×D60 [×3], D60⋊11C2 [×3], D4×D15 [×6], Q8⋊3D15 [×2], C15×C4○D4, D4⋊8D30
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], D15, 2+ (1+4), C22×D5 [×7], S3×C23, D30 [×7], C23×D5, D4○D12, C22×D15 [×7], D4⋊8D10, C23×D15, D4⋊8D30
Generators and relations
G = < a,b,c,d | a4=b2=c30=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >
►On 120 points
(1 106 89 31)(2 107 90 32)(3 108 61 33)(4 109 62 34)(5 110 63 35)(6 111 64 36)(7 112 65 37)(8 113 66 38)(9 114 67 39)(10 115 68 40)(11 116 69 41)(12 117 70 42)(13 118 71 43)(14 119 72 44)(15 120 73 45)(16 91 74 46)(17 92 75 47)(18 93 76 48)(19 94 77 49)(20 95 78 50)(21 96 79 51)(22 97 80 52)(23 98 81 53)(24 99 82 54)(25 100 83 55)(26 101 84 56)(27 102 85 57)(28 103 86 58)(29 104 87 59)(30 105 88 60)
(1 31)(2 107)(3 33)(4 109)(5 35)(6 111)(7 37)(8 113)(9 39)(10 115)(11 41)(12 117)(13 43)(14 119)(15 45)(16 91)(17 47)(18 93)(19 49)(20 95)(21 51)(22 97)(23 53)(24 99)(25 55)(26 101)(27 57)(28 103)(29 59)(30 105)(32 90)(34 62)(36 64)(38 66)(40 68)(42 70)(44 72)(46 74)(48 76)(50 78)(52 80)(54 82)(56 84)(58 86)(60 88)(61 108)(63 110)(65 112)(67 114)(69 116)(71 118)(73 120)(75 92)(77 94)(79 96)(81 98)(83 100)(85 102)(87 104)(89 106)
(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)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)
G:=sub<Sym(120)| (1,106,89,31)(2,107,90,32)(3,108,61,33)(4,109,62,34)(5,110,63,35)(6,111,64,36)(7,112,65,37)(8,113,66,38)(9,114,67,39)(10,115,68,40)(11,116,69,41)(12,117,70,42)(13,118,71,43)(14,119,72,44)(15,120,73,45)(16,91,74,46)(17,92,75,47)(18,93,76,48)(19,94,77,49)(20,95,78,50)(21,96,79,51)(22,97,80,52)(23,98,81,53)(24,99,82,54)(25,100,83,55)(26,101,84,56)(27,102,85,57)(28,103,86,58)(29,104,87,59)(30,105,88,60), (1,31)(2,107)(3,33)(4,109)(5,35)(6,111)(7,37)(8,113)(9,39)(10,115)(11,41)(12,117)(13,43)(14,119)(15,45)(16,91)(17,47)(18,93)(19,49)(20,95)(21,51)(22,97)(23,53)(24,99)(25,55)(26,101)(27,57)(28,103)(29,59)(30,105)(32,90)(34,62)(36,64)(38,66)(40,68)(42,70)(44,72)(46,74)(48,76)(50,78)(52,80)(54,82)(56,84)(58,86)(60,88)(61,108)(63,110)(65,112)(67,114)(69,116)(71,118)(73,120)(75,92)(77,94)(79,96)(81,98)(83,100)(85,102)(87,104)(89,106), (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)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,90)(88,89)>;
G:=Group( (1,106,89,31)(2,107,90,32)(3,108,61,33)(4,109,62,34)(5,110,63,35)(6,111,64,36)(7,112,65,37)(8,113,66,38)(9,114,67,39)(10,115,68,40)(11,116,69,41)(12,117,70,42)(13,118,71,43)(14,119,72,44)(15,120,73,45)(16,91,74,46)(17,92,75,47)(18,93,76,48)(19,94,77,49)(20,95,78,50)(21,96,79,51)(22,97,80,52)(23,98,81,53)(24,99,82,54)(25,100,83,55)(26,101,84,56)(27,102,85,57)(28,103,86,58)(29,104,87,59)(30,105,88,60), (1,31)(2,107)(3,33)(4,109)(5,35)(6,111)(7,37)(8,113)(9,39)(10,115)(11,41)(12,117)(13,43)(14,119)(15,45)(16,91)(17,47)(18,93)(19,49)(20,95)(21,51)(22,97)(23,53)(24,99)(25,55)(26,101)(27,57)(28,103)(29,59)(30,105)(32,90)(34,62)(36,64)(38,66)(40,68)(42,70)(44,72)(46,74)(48,76)(50,78)(52,80)(54,82)(56,84)(58,86)(60,88)(61,108)(63,110)(65,112)(67,114)(69,116)(71,118)(73,120)(75,92)(77,94)(79,96)(81,98)(83,100)(85,102)(87,104)(89,106), (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)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,90)(88,89) );
G=PermutationGroup([(1,106,89,31),(2,107,90,32),(3,108,61,33),(4,109,62,34),(5,110,63,35),(6,111,64,36),(7,112,65,37),(8,113,66,38),(9,114,67,39),(10,115,68,40),(11,116,69,41),(12,117,70,42),(13,118,71,43),(14,119,72,44),(15,120,73,45),(16,91,74,46),(17,92,75,47),(18,93,76,48),(19,94,77,49),(20,95,78,50),(21,96,79,51),(22,97,80,52),(23,98,81,53),(24,99,82,54),(25,100,83,55),(26,101,84,56),(27,102,85,57),(28,103,86,58),(29,104,87,59),(30,105,88,60)], [(1,31),(2,107),(3,33),(4,109),(5,35),(6,111),(7,37),(8,113),(9,39),(10,115),(11,41),(12,117),(13,43),(14,119),(15,45),(16,91),(17,47),(18,93),(19,49),(20,95),(21,51),(22,97),(23,53),(24,99),(25,55),(26,101),(27,57),(28,103),(29,59),(30,105),(32,90),(34,62),(36,64),(38,66),(40,68),(42,70),(44,72),(46,74),(48,76),(50,78),(52,80),(54,82),(56,84),(58,86),(60,88),(61,108),(63,110),(65,112),(67,114),(69,116),(71,118),(73,120),(75,92),(77,94),(79,96),(81,98),(83,100),(85,102),(87,104),(89,106)], [(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),(31,105),(32,104),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,96),(41,95),(42,94),(43,93),(44,92),(45,91),(46,120),(47,119),(48,118),(49,117),(50,116),(51,115),(52,114),(53,113),(54,112),(55,111),(56,110),(57,109),(58,108),(59,107),(60,106),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(87,90),(88,89)])
Matrix representation ►G ⊆ 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] >;
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 |
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