direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×2+ (1+4), C60.299C23, C30.102C24, C4○D4⋊5C30, (C2×D4)⋊6C30, D4⋊4(C2×C30), Q8⋊5(C2×C30), (C6×D4)⋊15C10, (D4×C30)⋊33C2, (D4×C10)⋊15C6, C23⋊2(C2×C30), (C2×C60)⋊39C22, C2.4(C23×C30), C4.9(C22×C30), (D4×C15)⋊44C22, C6.19(C23×C10), C10.19(C23×C6), C20.52(C22×C6), (C22×C30)⋊2C22, (Q8×C15)⋊39C22, C12.51(C22×C10), (C2×C30).263C23, C22.2(C22×C30), (C2×C20)⋊9(C2×C6), (C2×C4)⋊2(C2×C30), (C2×C12)⋊9(C2×C10), (C3×C4○D4)⋊8C10, (C5×C4○D4)⋊12C6, (C5×D4)⋊13(C2×C6), (C5×Q8)⋊14(C2×C6), (C15×C4○D4)⋊18C2, (C3×D4)⋊13(C2×C10), (C22×C6)⋊2(C2×C10), (C22×C10)⋊3(C2×C6), (C3×Q8)⋊12(C2×C10), (C2×C6).7(C22×C10), (C2×C10).7(C22×C6), SmallGroup(480,1184)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 440 in 332 conjugacy classes, 272 normal (12 characteristic)
C1, C2, C2 [×9], C3, C4 [×6], C22 [×9], C22 [×6], C5, C6, C6 [×9], C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C10, C10 [×9], C12 [×6], C2×C6 [×9], C2×C6 [×6], C15, C2×D4 [×9], C4○D4 [×6], C20 [×6], C2×C10 [×9], C2×C10 [×6], C2×C12 [×9], C3×D4 [×18], C3×Q8 [×2], C22×C6 [×6], C30, C30 [×9], 2+ (1+4), C2×C20 [×9], C5×D4 [×18], C5×Q8 [×2], C22×C10 [×6], C6×D4 [×9], C3×C4○D4 [×6], C60 [×6], C2×C30 [×9], C2×C30 [×6], D4×C10 [×9], C5×C4○D4 [×6], C3×2+ (1+4), C2×C60 [×9], D4×C15 [×18], Q8×C15 [×2], C22×C30 [×6], C5×2+ (1+4), D4×C30 [×9], C15×C4○D4 [×6], C15×2+ (1+4)
Quotients:
C1, C2 [×15], C3, C22 [×35], C5, C6 [×15], C23 [×15], C10 [×15], C2×C6 [×35], C15, C24, C2×C10 [×35], C22×C6 [×15], C30 [×15], 2+ (1+4), C22×C10 [×15], C23×C6, C2×C30 [×35], C23×C10, C3×2+ (1+4), C22×C30 [×15], C5×2+ (1+4), C23×C30, C15×2+ (1+4)
Generators and relations
G = < a,b,c,d,e | a15=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►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)
(1 113 50 74)(2 114 51 75)(3 115 52 61)(4 116 53 62)(5 117 54 63)(6 118 55 64)(7 119 56 65)(8 120 57 66)(9 106 58 67)(10 107 59 68)(11 108 60 69)(12 109 46 70)(13 110 47 71)(14 111 48 72)(15 112 49 73)(16 36 105 88)(17 37 91 89)(18 38 92 90)(19 39 93 76)(20 40 94 77)(21 41 95 78)(22 42 96 79)(23 43 97 80)(24 44 98 81)(25 45 99 82)(26 31 100 83)(27 32 101 84)(28 33 102 85)(29 34 103 86)(30 35 104 87)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 61)(43 62)(44 63)(45 64)(46 105)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)
(1 74 50 113)(2 75 51 114)(3 61 52 115)(4 62 53 116)(5 63 54 117)(6 64 55 118)(7 65 56 119)(8 66 57 120)(9 67 58 106)(10 68 59 107)(11 69 60 108)(12 70 46 109)(13 71 47 110)(14 72 48 111)(15 73 49 112)(16 36 105 88)(17 37 91 89)(18 38 92 90)(19 39 93 76)(20 40 94 77)(21 41 95 78)(22 42 96 79)(23 43 97 80)(24 44 98 81)(25 45 99 82)(26 31 100 83)(27 32 101 84)(28 33 102 85)(29 34 103 86)(30 35 104 87)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 76)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 106)(29 107)(30 108)(31 56)(32 57)(33 58)(34 59)(35 60)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 91)(72 92)(73 93)(74 94)(75 95)
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), (1,113,50,74)(2,114,51,75)(3,115,52,61)(4,116,53,62)(5,117,54,63)(6,118,55,64)(7,119,56,65)(8,120,57,66)(9,106,58,67)(10,107,59,68)(11,108,60,69)(12,109,46,70)(13,110,47,71)(14,111,48,72)(15,112,49,73)(16,36,105,88)(17,37,91,89)(18,38,92,90)(19,39,93,76)(20,40,94,77)(21,41,95,78)(22,42,96,79)(23,43,97,80)(24,44,98,81)(25,45,99,82)(26,31,100,83)(27,32,101,84)(28,33,102,85)(29,34,103,86)(30,35,104,87), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,61)(43,62)(44,63)(45,64)(46,105)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111), (1,74,50,113)(2,75,51,114)(3,61,52,115)(4,62,53,116)(5,63,54,117)(6,64,55,118)(7,65,56,119)(8,66,57,120)(9,67,58,106)(10,68,59,107)(11,69,60,108)(12,70,46,109)(13,71,47,110)(14,72,48,111)(15,73,49,112)(16,36,105,88)(17,37,91,89)(18,38,92,90)(19,39,93,76)(20,40,94,77)(21,41,95,78)(22,42,96,79)(23,43,97,80)(24,44,98,81)(25,45,99,82)(26,31,100,83)(27,32,101,84)(28,33,102,85)(29,34,103,86)(30,35,104,87), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,76)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,106)(29,107)(30,108)(31,56)(32,57)(33,58)(34,59)(35,60)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(61,96)(62,97)(63,98)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,91)(72,92)(73,93)(74,94)(75,95)>;
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), (1,113,50,74)(2,114,51,75)(3,115,52,61)(4,116,53,62)(5,117,54,63)(6,118,55,64)(7,119,56,65)(8,120,57,66)(9,106,58,67)(10,107,59,68)(11,108,60,69)(12,109,46,70)(13,110,47,71)(14,111,48,72)(15,112,49,73)(16,36,105,88)(17,37,91,89)(18,38,92,90)(19,39,93,76)(20,40,94,77)(21,41,95,78)(22,42,96,79)(23,43,97,80)(24,44,98,81)(25,45,99,82)(26,31,100,83)(27,32,101,84)(28,33,102,85)(29,34,103,86)(30,35,104,87), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,16)(13,17)(14,18)(15,19)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,61)(43,62)(44,63)(45,64)(46,105)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111), (1,74,50,113)(2,75,51,114)(3,61,52,115)(4,62,53,116)(5,63,54,117)(6,64,55,118)(7,65,56,119)(8,66,57,120)(9,67,58,106)(10,68,59,107)(11,69,60,108)(12,70,46,109)(13,71,47,110)(14,72,48,111)(15,73,49,112)(16,36,105,88)(17,37,91,89)(18,38,92,90)(19,39,93,76)(20,40,94,77)(21,41,95,78)(22,42,96,79)(23,43,97,80)(24,44,98,81)(25,45,99,82)(26,31,100,83)(27,32,101,84)(28,33,102,85)(29,34,103,86)(30,35,104,87), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,76)(16,109)(17,110)(18,111)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,106)(29,107)(30,108)(31,56)(32,57)(33,58)(34,59)(35,60)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(61,96)(62,97)(63,98)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,91)(72,92)(73,93)(74,94)(75,95) );
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)], [(1,113,50,74),(2,114,51,75),(3,115,52,61),(4,116,53,62),(5,117,54,63),(6,118,55,64),(7,119,56,65),(8,120,57,66),(9,106,58,67),(10,107,59,68),(11,108,60,69),(12,109,46,70),(13,110,47,71),(14,111,48,72),(15,112,49,73),(16,36,105,88),(17,37,91,89),(18,38,92,90),(19,39,93,76),(20,40,94,77),(21,41,95,78),(22,42,96,79),(23,43,97,80),(24,44,98,81),(25,45,99,82),(26,31,100,83),(27,32,101,84),(28,33,102,85),(29,34,103,86),(30,35,104,87)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,16),(13,17),(14,18),(15,19),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,75),(42,61),(43,62),(44,63),(45,64),(46,105),(47,91),(48,92),(49,93),(50,94),(51,95),(52,96),(53,97),(54,98),(55,99),(56,100),(57,101),(58,102),(59,103),(60,104),(76,112),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111)], [(1,74,50,113),(2,75,51,114),(3,61,52,115),(4,62,53,116),(5,63,54,117),(6,64,55,118),(7,65,56,119),(8,66,57,120),(9,67,58,106),(10,68,59,107),(11,69,60,108),(12,70,46,109),(13,71,47,110),(14,72,48,111),(15,73,49,112),(16,36,105,88),(17,37,91,89),(18,38,92,90),(19,39,93,76),(20,40,94,77),(21,41,95,78),(22,42,96,79),(23,43,97,80),(24,44,98,81),(25,45,99,82),(26,31,100,83),(27,32,101,84),(28,33,102,85),(29,34,103,86),(30,35,104,87)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,76),(16,109),(17,110),(18,111),(19,112),(20,113),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,106),(29,107),(30,108),(31,56),(32,57),(33,58),(34,59),(35,60),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(61,96),(62,97),(63,98),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105),(71,91),(72,92),(73,93),(74,94),(75,95)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 47 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 34 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 59 | 1 | 60 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 59 | 1 | 60 | 1 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 60 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 2 | 60 | 1 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 59 | 1 | 60 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(61))| [47,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[1,0,0,0,0,0,60,0,0,59,0,1,0,60,1,0,0,1,0,60,0,1,0,0,1],[1,0,0,0,0,0,60,59,0,0,0,0,1,0,0,0,0,60,0,1,0,0,1,1,0],[60,0,0,0,0,0,1,0,0,2,0,0,0,60,60,0,1,1,0,1,0,60,0,0,60],[1,0,0,0,0,0,1,0,59,0,0,0,0,1,1,0,0,0,60,0,0,0,1,1,0] >;
255 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2J | 3A | 3B | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | ··· | 6T | 10A | 10B | 10C | 10D | 10E | ··· | 10AN | 12A | ··· | 12L | 15A | ··· | 15H | 20A | ··· | 20X | 30A | ··· | 30H | 30I | ··· | 30CB | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
255 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | 2+ (1+4) | C3×2+ (1+4) | C5×2+ (1+4) | C15×2+ (1+4) |
| kernel | C15×2+ (1+4) | D4×C30 | C15×C4○D4 | C5×2+ (1+4) | C3×2+ (1+4) | D4×C10 | C5×C4○D4 | C6×D4 | C3×C4○D4 | 2+ (1+4) | C2×D4 | C4○D4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 9 | 6 | 2 | 4 | 18 | 12 | 36 | 24 | 8 | 72 | 48 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_{15}\times 2_+^{(1+4)} % in TeX
G:=Group("C15xES+(2,2)"); // GroupNames label
G:=SmallGroup(480,1184);
// by ID
G=gap.SmallGroup(480,1184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389,2571,6947]);
// Polycyclic
G:=Group<a,b,c,d,e|a^15=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀