direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4⋊8D10, C30.80C24, C60.215C23, C15⋊132+ (1+4), D4⋊8(C6×D5), (D4×D5)⋊5C6, Q8⋊8(C6×D5), C4○D20⋊8C6, (C2×D20)⋊13C6, (C6×D20)⋊29C2, (C3×D4)⋊30D10, D20⋊12(C2×C6), (C2×C12)⋊23D10, Q8⋊2D5⋊8C6, (C3×Q8)⋊26D10, (C2×C60)⋊24C22, C6.80(C23×D5), Dic10⋊12(C2×C6), C5⋊2(C3×2+ (1+4)), (D4×C15)⋊33C22, (C3×D20)⋊41C22, (D5×C12)⋊15C22, C10.12(C23×C6), C20.25(C22×C6), (Q8×C15)⋊29C22, D10.6(C22×C6), (C6×D5).58C23, (C2×C30).256C23, C12.215(C22×D5), Dic5.7(C22×C6), (C3×Dic10)⋊39C22, (C3×Dic5).60C23, (C3×D4×D5)⋊12C2, (C2×C4)⋊4(C6×D5), C4.33(D5×C2×C6), (C2×C20)⋊5(C2×C6), C4○D4⋊5(C3×D5), (C3×C4○D4)⋊8D5, (C5×C4○D4)⋊8C6, (C4×D5)⋊2(C2×C6), (C5×D4)⋊9(C2×C6), C5⋊D4⋊5(C2×C6), (C15×C4○D4)⋊9C2, C22.3(D5×C2×C6), (C5×Q8)⋊10(C2×C6), (D5×C2×C6)⋊16C22, (C3×C4○D20)⋊18C2, C2.13(D5×C22×C6), (C22×D5)⋊4(C2×C6), (C3×Q8⋊2D5)⋊12C2, (C3×C5⋊D4)⋊20C22, (C2×C10).4(C22×C6), (C2×C6).23(C22×D5), SmallGroup(480,1146)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1136 in 332 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, C6, C6 [×9], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], D5 [×6], C10, C10 [×3], C12, C12 [×3], C12 [×2], C2×C6 [×3], C2×C6 [×12], C15, C2×D4 [×9], C4○D4, C4○D4 [×5], Dic5 [×2], C20, C20 [×3], D10 [×6], D10 [×6], C2×C10 [×3], C2×C12 [×3], C2×C12 [×6], C3×D4 [×3], C3×D4 [×15], C3×Q8, C3×Q8, C22×C6 [×6], C3×D5 [×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], C6×D4 [×9], C3×C4○D4, C3×C4○D4 [×5], C3×Dic5 [×2], C60, C60 [×3], C6×D5 [×6], C6×D5 [×6], C2×C30 [×3], C2×D20 [×3], C4○D20 [×3], D4×D5 [×6], Q8⋊2D5 [×2], C5×C4○D4, C3×2+ (1+4), C3×Dic10, D5×C12 [×6], C3×D20 [×9], C3×C5⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, D5×C2×C6 [×6], D4⋊8D10, C6×D20 [×3], C3×C4○D20 [×3], C3×D4×D5 [×6], C3×Q8⋊2D5 [×2], C15×C4○D4, C3×D4⋊8D10
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C24, D10 [×7], C22×C6 [×15], C3×D5, 2+ (1+4), C22×D5 [×7], C23×C6, C6×D5 [×7], C23×D5, C3×2+ (1+4), D5×C2×C6 [×7], D4⋊8D10, D5×C22×C6, C3×D4⋊8D10
Generators and relations
G = < a,b,c,d,e | a3=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ce=ec, ede=d-1 >
►On 120 points
Generators in S120
(1 22 12)(2 23 13)(3 24 14)(4 25 15)(5 21 11)(6 26 17)(7 27 18)(8 28 19)(9 29 20)(10 30 16)(31 55 45)(32 51 41)(33 52 42)(34 53 43)(35 54 44)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 88 77)(62 89 78)(63 90 79)(64 81 80)(65 82 71)(66 83 72)(67 84 73)(68 85 74)(69 86 75)(70 87 76)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(1 36 7 32)(2 37 8 33)(3 38 9 34)(4 39 10 35)(5 40 6 31)(11 50 17 45)(12 46 18 41)(13 47 19 42)(14 48 20 43)(15 49 16 44)(21 60 26 55)(22 56 27 51)(23 57 28 52)(24 58 29 53)(25 59 30 54)(61 97 66 92)(62 98 67 93)(63 99 68 94)(64 100 69 95)(65 91 70 96)(71 101 76 106)(72 102 77 107)(73 103 78 108)(74 104 79 109)(75 105 80 110)(81 120 86 115)(82 111 87 116)(83 112 88 117)(84 113 89 118)(85 114 90 119)
(1 92)(2 98)(3 94)(4 100)(5 96)(6 91)(7 97)(8 93)(9 99)(10 95)(11 106)(12 102)(13 108)(14 104)(15 110)(16 105)(17 101)(18 107)(19 103)(20 109)(21 116)(22 112)(23 118)(24 114)(25 120)(26 111)(27 117)(28 113)(29 119)(30 115)(31 65)(32 61)(33 67)(34 63)(35 69)(36 66)(37 62)(38 68)(39 64)(40 70)(41 77)(42 73)(43 79)(44 75)(45 71)(46 72)(47 78)(48 74)(49 80)(50 76)(51 88)(52 84)(53 90)(54 86)(55 82)(56 83)(57 89)(58 85)(59 81)(60 87)
(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 5)(2 4)(6 7)(8 10)(11 12)(13 15)(16 19)(17 18)(21 22)(23 25)(26 27)(28 30)(31 36)(32 40)(33 39)(34 38)(35 37)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)(85 90)(86 89)(87 88)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)
G:=sub<Sym(120)| (1,22,12)(2,23,13)(3,24,14)(4,25,15)(5,21,11)(6,26,17)(7,27,18)(8,28,19)(9,29,20)(10,30,16)(31,55,45)(32,51,41)(33,52,42)(34,53,43)(35,54,44)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,88,77)(62,89,78)(63,90,79)(64,81,80)(65,82,71)(66,83,72)(67,84,73)(68,85,74)(69,86,75)(70,87,76)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,36,7,32)(2,37,8,33)(3,38,9,34)(4,39,10,35)(5,40,6,31)(11,50,17,45)(12,46,18,41)(13,47,19,42)(14,48,20,43)(15,49,16,44)(21,60,26,55)(22,56,27,51)(23,57,28,52)(24,58,29,53)(25,59,30,54)(61,97,66,92)(62,98,67,93)(63,99,68,94)(64,100,69,95)(65,91,70,96)(71,101,76,106)(72,102,77,107)(73,103,78,108)(74,104,79,109)(75,105,80,110)(81,120,86,115)(82,111,87,116)(83,112,88,117)(84,113,89,118)(85,114,90,119), (1,92)(2,98)(3,94)(4,100)(5,96)(6,91)(7,97)(8,93)(9,99)(10,95)(11,106)(12,102)(13,108)(14,104)(15,110)(16,105)(17,101)(18,107)(19,103)(20,109)(21,116)(22,112)(23,118)(24,114)(25,120)(26,111)(27,117)(28,113)(29,119)(30,115)(31,65)(32,61)(33,67)(34,63)(35,69)(36,66)(37,62)(38,68)(39,64)(40,70)(41,77)(42,73)(43,79)(44,75)(45,71)(46,72)(47,78)(48,74)(49,80)(50,76)(51,88)(52,84)(53,90)(54,86)(55,82)(56,83)(57,89)(58,85)(59,81)(60,87), (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), (1,5)(2,4)(6,7)(8,10)(11,12)(13,15)(16,19)(17,18)(21,22)(23,25)(26,27)(28,30)(31,36)(32,40)(33,39)(34,38)(35,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)(85,90)(86,89)(87,88)(91,97)(92,96)(93,95)(98,100)(101,107)(102,106)(103,105)(108,110)(111,117)(112,116)(113,115)(118,120)>;
G:=Group( (1,22,12)(2,23,13)(3,24,14)(4,25,15)(5,21,11)(6,26,17)(7,27,18)(8,28,19)(9,29,20)(10,30,16)(31,55,45)(32,51,41)(33,52,42)(34,53,43)(35,54,44)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,88,77)(62,89,78)(63,90,79)(64,81,80)(65,82,71)(66,83,72)(67,84,73)(68,85,74)(69,86,75)(70,87,76)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,36,7,32)(2,37,8,33)(3,38,9,34)(4,39,10,35)(5,40,6,31)(11,50,17,45)(12,46,18,41)(13,47,19,42)(14,48,20,43)(15,49,16,44)(21,60,26,55)(22,56,27,51)(23,57,28,52)(24,58,29,53)(25,59,30,54)(61,97,66,92)(62,98,67,93)(63,99,68,94)(64,100,69,95)(65,91,70,96)(71,101,76,106)(72,102,77,107)(73,103,78,108)(74,104,79,109)(75,105,80,110)(81,120,86,115)(82,111,87,116)(83,112,88,117)(84,113,89,118)(85,114,90,119), (1,92)(2,98)(3,94)(4,100)(5,96)(6,91)(7,97)(8,93)(9,99)(10,95)(11,106)(12,102)(13,108)(14,104)(15,110)(16,105)(17,101)(18,107)(19,103)(20,109)(21,116)(22,112)(23,118)(24,114)(25,120)(26,111)(27,117)(28,113)(29,119)(30,115)(31,65)(32,61)(33,67)(34,63)(35,69)(36,66)(37,62)(38,68)(39,64)(40,70)(41,77)(42,73)(43,79)(44,75)(45,71)(46,72)(47,78)(48,74)(49,80)(50,76)(51,88)(52,84)(53,90)(54,86)(55,82)(56,83)(57,89)(58,85)(59,81)(60,87), (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), (1,5)(2,4)(6,7)(8,10)(11,12)(13,15)(16,19)(17,18)(21,22)(23,25)(26,27)(28,30)(31,36)(32,40)(33,39)(34,38)(35,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)(85,90)(86,89)(87,88)(91,97)(92,96)(93,95)(98,100)(101,107)(102,106)(103,105)(108,110)(111,117)(112,116)(113,115)(118,120) );
G=PermutationGroup([(1,22,12),(2,23,13),(3,24,14),(4,25,15),(5,21,11),(6,26,17),(7,27,18),(8,28,19),(9,29,20),(10,30,16),(31,55,45),(32,51,41),(33,52,42),(34,53,43),(35,54,44),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,88,77),(62,89,78),(63,90,79),(64,81,80),(65,82,71),(66,83,72),(67,84,73),(68,85,74),(69,86,75),(70,87,76),(91,111,101),(92,112,102),(93,113,103),(94,114,104),(95,115,105),(96,116,106),(97,117,107),(98,118,108),(99,119,109),(100,120,110)], [(1,36,7,32),(2,37,8,33),(3,38,9,34),(4,39,10,35),(5,40,6,31),(11,50,17,45),(12,46,18,41),(13,47,19,42),(14,48,20,43),(15,49,16,44),(21,60,26,55),(22,56,27,51),(23,57,28,52),(24,58,29,53),(25,59,30,54),(61,97,66,92),(62,98,67,93),(63,99,68,94),(64,100,69,95),(65,91,70,96),(71,101,76,106),(72,102,77,107),(73,103,78,108),(74,104,79,109),(75,105,80,110),(81,120,86,115),(82,111,87,116),(83,112,88,117),(84,113,89,118),(85,114,90,119)], [(1,92),(2,98),(3,94),(4,100),(5,96),(6,91),(7,97),(8,93),(9,99),(10,95),(11,106),(12,102),(13,108),(14,104),(15,110),(16,105),(17,101),(18,107),(19,103),(20,109),(21,116),(22,112),(23,118),(24,114),(25,120),(26,111),(27,117),(28,113),(29,119),(30,115),(31,65),(32,61),(33,67),(34,63),(35,69),(36,66),(37,62),(38,68),(39,64),(40,70),(41,77),(42,73),(43,79),(44,75),(45,71),(46,72),(47,78),(48,74),(49,80),(50,76),(51,88),(52,84),(53,90),(54,86),(55,82),(56,83),(57,89),(58,85),(59,81),(60,87)], [(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)], [(1,5),(2,4),(6,7),(8,10),(11,12),(13,15),(16,19),(17,18),(21,22),(23,25),(26,27),(28,30),(31,36),(32,40),(33,39),(34,38),(35,37),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,70),(62,69),(63,68),(64,67),(65,66),(71,72),(73,80),(74,79),(75,78),(76,77),(81,84),(82,83),(85,90),(86,89),(87,88),(91,97),(92,96),(93,95),(98,100),(101,107),(102,106),(103,105),(108,110),(111,117),(112,116),(113,115),(118,120)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 25 | 54 | 0 | 0 |
| 11 | 36 | 0 | 0 |
| 16 | 40 | 32 | 7 |
| 16 | 37 | 54 | 29 |
| 36 | 7 | 60 | 39 |
| 50 | 25 | 40 | 21 |
| 45 | 21 | 29 | 54 |
| 45 | 24 | 7 | 32 |
| 0 | 17 | 0 | 0 |
| 43 | 43 | 0 | 0 |
| 53 | 47 | 1 | 44 |
| 0 | 6 | 17 | 17 |
| 1 | 0 | 0 | 0 |
| 42 | 60 | 0 | 0 |
| 53 | 47 | 1 | 44 |
| 47 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[25,11,16,16,54,36,40,37,0,0,32,54,0,0,7,29],[36,50,45,45,7,25,21,24,60,40,29,7,39,21,54,32],[0,43,53,0,17,43,47,6,0,0,1,17,0,0,44,17],[1,42,53,47,0,60,47,0,0,0,1,0,0,0,44,60] >;
111 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6T | 10A | 10B | 10C | ··· | 10H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 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 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 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 | 10 | ··· | 10 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D5 | D10 | D10 | D10 | C3×D5 | C6×D5 | C6×D5 | C6×D5 | 2+ (1+4) | C3×2+ (1+4) | D4⋊8D10 | C3×D4⋊8D10 |
| kernel | C3×D4⋊8D10 | C6×D20 | C3×C4○D20 | C3×D4×D5 | C3×Q8⋊2D5 | C15×C4○D4 | D4⋊8D10 | C2×D20 | C4○D20 | D4×D5 | Q8⋊2D5 | C5×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 2 | 6 | 6 | 12 | 4 | 2 | 2 | 6 | 6 | 2 | 4 | 12 | 12 | 4 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_8D_{10} % in TeX
G:=Group("C3xD4:8D10"); // GroupNames label
G:=SmallGroup(480,1146);
// by ID
G=gap.SmallGroup(480,1146);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,555,1571,192,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^10=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀