metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊8C22, Dic6⋊12D10, Dic10⋊12D6, C30.28C24, C60.52C23, D30.13C23, Dic15.15C23, (C4×D5)⋊9D6, D4⋊9(S3×D5), C5⋊D4⋊4D6, (C4×S3)⋊9D10, (C5×D4)⋊12D6, (D4×D15)⋊5C2, C3⋊D4⋊4D10, D15⋊Q8⋊5C2, C15⋊Q8⋊3C22, D4⋊2S3⋊6D5, D4⋊2D5⋊6S3, (C3×D4)⋊12D10, D15⋊2(C4○D4), C12.28D10⋊4C2, D10⋊D6⋊3C2, D60⋊C2⋊4C2, (S3×C20)⋊3C22, (C2×Dic5)⋊15D6, (D5×C12)⋊3C22, C5⋊D12⋊5C22, C15⋊7D4⋊4C22, C3⋊D20⋊5C22, (C2×C30).4C23, C6.28(C23×D5), (C2×Dic3)⋊15D10, Dic3.D10⋊4C2, Dic5.D6⋊4C2, (D4×C15)⋊10C22, C10.28(S3×C23), C20.52(C22×S3), (C5×Dic6)⋊8C22, (C6×D5).12C23, D6.13(C22×D5), C12.52(C22×D5), (S3×C10).13C23, D30.C2⋊12C22, (S3×Dic5)⋊12C22, (D5×Dic3)⋊12C22, (C3×Dic10)⋊8C22, (C6×Dic5)⋊13C22, (C4×D15).18C22, D10.13(C22×S3), (C10×Dic3)⋊13C22, Dic3.14(C22×D5), (C5×Dic3).15C23, Dic5.14(C22×S3), (C3×Dic5).13C23, (C22×D15).74C22, (C4×S3×D5)⋊4C2, C5⋊4(S3×C4○D4), C3⋊4(D5×C4○D4), C4.52(C2×S3×D5), C15⋊13(C2×C4○D4), C22.4(C2×S3×D5), (C3×D4⋊2D5)⋊6C2, (C5×D4⋊2S3)⋊6C2, C2.31(C22×S3×D5), (C3×C5⋊D4)⋊4C22, (C5×C3⋊D4)⋊4C22, (C2×S3×D5).10C22, (C2×C6).4(C22×D5), (C2×D30.C2)⋊21C2, (C2×C10).4(C22×S3), SmallGroup(480,1100)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1692 in 328 conjugacy classes, 110 normal (50 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×7], C22 [×2], C22 [×11], C5, S3 [×5], C6, C6 [×3], C2×C4 [×16], D4, D4 [×11], Q8 [×4], C23 [×3], D5 [×5], C10, C10 [×3], Dic3, Dic3 [×2], Dic3, C12, C12 [×3], D6, D6 [×9], C2×C6 [×2], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5, Dic5 [×2], Dic5, C20, C20 [×3], D10, D10 [×9], C2×C10 [×2], C2×C10, Dic6, Dic6 [×2], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×3], C5×S3, C3×D5, D15 [×2], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10, Dic10 [×2], C4×D5, C4×D5 [×9], D20 [×3], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D4⋊2S3, D4⋊2S3 [×2], S3×Q8, Q8⋊3S3, C3×C4○D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, C3×Dic5 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, D30 [×2], D30 [×4], C2×C30 [×2], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D4⋊2D5, D4⋊2D5 [×2], Q8×D5, Q8⋊2D5, C5×C4○D4, S3×C4○D4, D5×Dic3, S3×Dic5, D30.C2, D30.C2 [×6], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], C4×D15, D60, C15⋊7D4 [×2], D4×C15, C2×S3×D5, C22×D15 [×2], D5×C4○D4, D60⋊C2, D15⋊Q8, C12.28D10, C4×S3×D5, Dic5.D6 [×2], Dic3.D10 [×2], C2×D30.C2 [×2], D10⋊D6 [×2], C3×D4⋊2D5, C5×D4⋊2S3, D4×D15, D30.C23
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D30.C23
Generators and relations
G = < a,b,c,d,e | a30=b2=c2=e2=1, d2=a15, bab=eae=a-1, cac=a11, ad=da, cbc=a25b, dbd-1=a15b, ebe=a28b, cd=dc, ce=ec, de=ed >
►On 120 points
(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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 120)(87 119)(88 118)(89 117)(90 116)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 46)(32 57)(33 38)(34 49)(35 60)(36 41)(37 52)(39 44)(40 55)(42 47)(43 58)(45 50)(48 53)(51 56)(54 59)(61 81)(63 73)(64 84)(66 76)(67 87)(69 79)(70 90)(72 82)(75 85)(78 88)(91 106)(92 117)(93 98)(94 109)(95 120)(96 101)(97 112)(99 104)(100 115)(102 107)(103 118)(105 110)(108 113)(111 116)(114 119)
(1 71 16 86)(2 72 17 87)(3 73 18 88)(4 74 19 89)(5 75 20 90)(6 76 21 61)(7 77 22 62)(8 78 23 63)(9 79 24 64)(10 80 25 65)(11 81 26 66)(12 82 27 67)(13 83 28 68)(14 84 29 69)(15 85 30 70)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(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 66)(62 65)(63 64)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,120)(87,119)(88,118)(89,117)(90,116), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,46)(32,57)(33,38)(34,49)(35,60)(36,41)(37,52)(39,44)(40,55)(42,47)(43,58)(45,50)(48,53)(51,56)(54,59)(61,81)(63,73)(64,84)(66,76)(67,87)(69,79)(70,90)(72,82)(75,85)(78,88)(91,106)(92,117)(93,98)(94,109)(95,120)(96,101)(97,112)(99,104)(100,115)(102,107)(103,118)(105,110)(108,113)(111,116)(114,119), (1,71,16,86)(2,72,17,87)(3,73,18,88)(4,74,19,89)(5,75,20,90)(6,76,21,61)(7,77,22,62)(8,78,23,63)(9,79,24,64)(10,80,25,65)(11,81,26,66)(12,82,27,67)(13,83,28,68)(14,84,29,69)(15,85,30,70)(31,106,46,91)(32,107,47,92)(33,108,48,93)(34,109,49,94)(35,110,50,95)(36,111,51,96)(37,112,52,97)(38,113,53,98)(39,114,54,99)(40,115,55,100)(41,116,56,101)(42,117,57,102)(43,118,58,103)(44,119,59,104)(45,120,60,105), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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)(61,66)(62,65)(63,64)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)>;
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,120)(87,119)(88,118)(89,117)(90,116), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,46)(32,57)(33,38)(34,49)(35,60)(36,41)(37,52)(39,44)(40,55)(42,47)(43,58)(45,50)(48,53)(51,56)(54,59)(61,81)(63,73)(64,84)(66,76)(67,87)(69,79)(70,90)(72,82)(75,85)(78,88)(91,106)(92,117)(93,98)(94,109)(95,120)(96,101)(97,112)(99,104)(100,115)(102,107)(103,118)(105,110)(108,113)(111,116)(114,119), (1,71,16,86)(2,72,17,87)(3,73,18,88)(4,74,19,89)(5,75,20,90)(6,76,21,61)(7,77,22,62)(8,78,23,63)(9,79,24,64)(10,80,25,65)(11,81,26,66)(12,82,27,67)(13,83,28,68)(14,84,29,69)(15,85,30,70)(31,106,46,91)(32,107,47,92)(33,108,48,93)(34,109,49,94)(35,110,50,95)(36,111,51,96)(37,112,52,97)(38,113,53,98)(39,114,54,99)(40,115,55,100)(41,116,56,101)(42,117,57,102)(43,118,58,103)(44,119,59,104)(45,120,60,105), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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)(61,66)(62,65)(63,64)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114) );
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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,60),(17,59),(18,58),(19,57),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(61,115),(62,114),(63,113),(64,112),(65,111),(66,110),(67,109),(68,108),(69,107),(70,106),(71,105),(72,104),(73,103),(74,102),(75,101),(76,100),(77,99),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(85,91),(86,120),(87,119),(88,118),(89,117),(90,116)], [(2,12),(3,23),(5,15),(6,26),(8,18),(9,29),(11,21),(14,24),(17,27),(20,30),(31,46),(32,57),(33,38),(34,49),(35,60),(36,41),(37,52),(39,44),(40,55),(42,47),(43,58),(45,50),(48,53),(51,56),(54,59),(61,81),(63,73),(64,84),(66,76),(67,87),(69,79),(70,90),(72,82),(75,85),(78,88),(91,106),(92,117),(93,98),(94,109),(95,120),(96,101),(97,112),(99,104),(100,115),(102,107),(103,118),(105,110),(108,113),(111,116),(114,119)], [(1,71,16,86),(2,72,17,87),(3,73,18,88),(4,74,19,89),(5,75,20,90),(6,76,21,61),(7,77,22,62),(8,78,23,63),(9,79,24,64),(10,80,25,65),(11,81,26,66),(12,82,27,67),(13,83,28,68),(14,84,29,69),(15,85,30,70),(31,106,46,91),(32,107,47,92),(33,108,48,93),(34,109,49,94),(35,110,50,95),(36,111,51,96),(37,112,52,97),(38,113,53,98),(39,114,54,99),(40,115,55,100),(41,116,56,101),(42,117,57,102),(43,118,58,103),(44,119,59,104),(45,120,60,105)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(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),(61,66),(62,65),(63,64),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 60 | 46 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 34 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,46,1,0,0,0,0,0,0,43,18,0,0,0,0,60,18,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[1,8,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[50,34,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 10 | 15 | 15 | 30 | 30 | 2 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 10 | 10 | 30 | 2 | 2 | 2 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | D10 | S3×D5 | S3×C4○D4 | C2×S3×D5 | C2×S3×D5 | D5×C4○D4 | D30.C23 |
| kernel | D30.C23 | D60⋊C2 | D15⋊Q8 | C12.28D10 | C4×S3×D5 | Dic5.D6 | Dic3.D10 | C2×D30.C2 | D10⋊D6 | C3×D4⋊2D5 | C5×D4⋊2S3 | D4×D15 | D4⋊2D5 | D4⋊2S3 | Dic10 | C4×D5 | C2×Dic5 | C5⋊D4 | C5×D4 | D15 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | D4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
In GAP, Magma, Sage, TeX
D_{30}.C_2^3 % in TeX
G:=Group("D30.C2^3"); // GroupNames label
G:=SmallGroup(480,1100);
// by ID
G=gap.SmallGroup(480,1100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,185,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^30=b^2=c^2=e^2=1,d^2=a^15,b*a*b=e*a*e=a^-1,c*a*c=a^11,a*d=d*a,c*b*c=a^25*b,d*b*d^-1=a^15*b,e*b*e=a^28*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations