direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4⋊6D6, C30.90C24, C60.237C23, C15⋊112+ (1+4), (S3×D4)⋊4C10, D4⋊6(S3×C10), (C6×D4)⋊7C10, (C5×D4)⋊28D6, (C2×C20)⋊22D6, C4○D12⋊5C10, (D4×C30)⋊21C2, D12⋊8(C2×C10), (D4×C10)⋊16S3, C23⋊3(S3×C10), (C22×C10)⋊5D6, D4⋊2S3⋊4C10, (C2×C60)⋊28C22, Dic6⋊8(C2×C10), C6.7(C23×C10), (S3×C20)⋊14C22, C3⋊1(C5×2+ (1+4)), (C5×D12)⋊38C22, (D4×C15)⋊38C22, C10.75(S3×C23), D6.3(C22×C10), (S3×C10).38C23, C12.21(C22×C10), (C2×C30).258C23, C20.210(C22×S3), (C22×C30)⋊17C22, (C5×Dic6)⋊35C22, (C10×Dic3)⋊21C22, (C5×Dic3).40C23, Dic3.4(C22×C10), (C5×S3×D4)⋊11C2, (C2×C4)⋊3(S3×C10), (C2×D4)⋊7(C5×S3), C4.21(S3×C2×C10), (C4×S3)⋊1(C2×C10), (C2×C12)⋊3(C2×C10), (C3×D4)⋊7(C2×C10), C3⋊D4⋊3(C2×C10), C22.2(S3×C2×C10), C2.8(S3×C22×C10), (C5×C4○D12)⋊15C2, (C10×C3⋊D4)⋊26C2, (C2×C3⋊D4)⋊11C10, (S3×C2×C10)⋊15C22, (C22×C6)⋊5(C2×C10), (C5×D4⋊2S3)⋊11C2, (C22×S3)⋊3(C2×C10), (C2×Dic3)⋊4(C2×C10), (C5×C3⋊D4)⋊19C22, (C2×C6).2(C22×C10), (C2×C10).258(C22×S3), SmallGroup(480,1156)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 740 in 332 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C5, S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C10, C10 [×9], Dic3 [×4], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×2], C15, C2×D4, C2×D4 [×8], C4○D4 [×6], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], C2×C10 [×10], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], C5×S3 [×4], C30, C30 [×5], 2+ (1+4), C2×C20, C2×C20 [×8], C5×D4 [×4], C5×D4 [×14], C5×Q8 [×2], C22×C10 [×2], C22×C10 [×4], C4○D12 [×2], S3×D4 [×4], D4⋊2S3 [×4], C2×C3⋊D4 [×4], C6×D4, C5×Dic3 [×4], C60 [×2], S3×C10 [×4], S3×C10 [×4], C2×C30, C2×C30 [×4], C2×C30 [×2], D4×C10, D4×C10 [×8], C5×C4○D4 [×6], D4⋊6D6, C5×Dic6 [×2], S3×C20 [×4], C5×D12 [×2], C10×Dic3 [×4], C5×C3⋊D4 [×12], C2×C60, D4×C15 [×4], S3×C2×C10 [×4], C22×C30 [×2], C5×2+ (1+4), C5×C4○D12 [×2], C5×S3×D4 [×4], C5×D4⋊2S3 [×4], C10×C3⋊D4 [×4], D4×C30, C5×D4⋊6D6
Quotients:
C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, 2+ (1+4), C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, D4⋊6D6, S3×C2×C10 [×7], C5×2+ (1+4), S3×C22×C10, C5×D4⋊6D6
Generators and relations
G = < a,b,c,d,e | a5=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 120 points
(1 25 19 13 8)(2 26 20 14 9)(3 27 21 15 7)(4 30 24 18 10)(5 28 22 16 11)(6 29 23 17 12)(31 57 49 43 37)(32 55 50 44 38)(33 56 51 45 39)(34 59 52 46 40)(35 60 53 47 41)(36 58 54 48 42)(61 89 83 77 71)(62 90 84 78 72)(63 85 79 73 67)(64 86 80 74 68)(65 87 81 75 69)(66 88 82 76 70)(91 115 109 103 97)(92 116 110 104 98)(93 117 111 105 99)(94 118 112 106 100)(95 119 113 107 101)(96 120 114 108 102)
(1 92 5 95)(2 96 6 93)(3 94 4 91)(7 100 10 97)(8 98 11 101)(9 102 12 99)(13 104 16 107)(14 108 17 105)(15 106 18 103)(19 110 22 113)(20 114 23 111)(21 112 24 109)(25 116 28 119)(26 120 29 117)(27 118 30 115)(31 66 35 63)(32 64 36 61)(33 62 34 65)(37 70 41 67)(38 68 42 71)(39 72 40 69)(43 76 47 73)(44 74 48 77)(45 78 46 75)(49 82 53 79)(50 80 54 83)(51 84 52 81)(55 86 58 89)(56 90 59 87)(57 88 60 85)
(1 61)(2 65)(3 63)(4 66)(5 64)(6 62)(7 67)(8 71)(9 69)(10 70)(11 68)(12 72)(13 77)(14 75)(15 73)(16 74)(17 78)(18 76)(19 83)(20 81)(21 79)(22 80)(23 84)(24 82)(25 89)(26 87)(27 85)(28 86)(29 90)(30 88)(31 91)(32 95)(33 93)(34 96)(35 94)(36 92)(37 97)(38 101)(39 99)(40 102)(41 100)(42 98)(43 103)(44 107)(45 105)(46 108)(47 106)(48 104)(49 109)(50 113)(51 111)(52 114)(53 112)(54 110)(55 119)(56 117)(57 115)(58 116)(59 120)(60 118)
(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 3)(4 5)(7 8)(10 11)(13 15)(16 18)(19 21)(22 24)(25 27)(28 30)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)(61 66)(62 65)(63 64)(67 68)(69 72)(70 71)(73 74)(75 78)(76 77)(79 80)(81 84)(82 83)(85 86)(87 90)(88 89)(91 95)(92 94)(97 101)(98 100)(103 107)(104 106)(109 113)(110 112)(115 119)(116 118)
G:=sub<Sym(120)| (1,25,19,13,8)(2,26,20,14,9)(3,27,21,15,7)(4,30,24,18,10)(5,28,22,16,11)(6,29,23,17,12)(31,57,49,43,37)(32,55,50,44,38)(33,56,51,45,39)(34,59,52,46,40)(35,60,53,47,41)(36,58,54,48,42)(61,89,83,77,71)(62,90,84,78,72)(63,85,79,73,67)(64,86,80,74,68)(65,87,81,75,69)(66,88,82,76,70)(91,115,109,103,97)(92,116,110,104,98)(93,117,111,105,99)(94,118,112,106,100)(95,119,113,107,101)(96,120,114,108,102), (1,92,5,95)(2,96,6,93)(3,94,4,91)(7,100,10,97)(8,98,11,101)(9,102,12,99)(13,104,16,107)(14,108,17,105)(15,106,18,103)(19,110,22,113)(20,114,23,111)(21,112,24,109)(25,116,28,119)(26,120,29,117)(27,118,30,115)(31,66,35,63)(32,64,36,61)(33,62,34,65)(37,70,41,67)(38,68,42,71)(39,72,40,69)(43,76,47,73)(44,74,48,77)(45,78,46,75)(49,82,53,79)(50,80,54,83)(51,84,52,81)(55,86,58,89)(56,90,59,87)(57,88,60,85), (1,61)(2,65)(3,63)(4,66)(5,64)(6,62)(7,67)(8,71)(9,69)(10,70)(11,68)(12,72)(13,77)(14,75)(15,73)(16,74)(17,78)(18,76)(19,83)(20,81)(21,79)(22,80)(23,84)(24,82)(25,89)(26,87)(27,85)(28,86)(29,90)(30,88)(31,91)(32,95)(33,93)(34,96)(35,94)(36,92)(37,97)(38,101)(39,99)(40,102)(41,100)(42,98)(43,103)(44,107)(45,105)(46,108)(47,106)(48,104)(49,109)(50,113)(51,111)(52,114)(53,112)(54,110)(55,119)(56,117)(57,115)(58,116)(59,120)(60,118), (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), (1,3)(4,5)(7,8)(10,11)(13,15)(16,18)(19,21)(22,24)(25,27)(28,30)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40)(43,48)(44,47)(45,46)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,68)(69,72)(70,71)(73,74)(75,78)(76,77)(79,80)(81,84)(82,83)(85,86)(87,90)(88,89)(91,95)(92,94)(97,101)(98,100)(103,107)(104,106)(109,113)(110,112)(115,119)(116,118)>;
G:=Group( (1,25,19,13,8)(2,26,20,14,9)(3,27,21,15,7)(4,30,24,18,10)(5,28,22,16,11)(6,29,23,17,12)(31,57,49,43,37)(32,55,50,44,38)(33,56,51,45,39)(34,59,52,46,40)(35,60,53,47,41)(36,58,54,48,42)(61,89,83,77,71)(62,90,84,78,72)(63,85,79,73,67)(64,86,80,74,68)(65,87,81,75,69)(66,88,82,76,70)(91,115,109,103,97)(92,116,110,104,98)(93,117,111,105,99)(94,118,112,106,100)(95,119,113,107,101)(96,120,114,108,102), (1,92,5,95)(2,96,6,93)(3,94,4,91)(7,100,10,97)(8,98,11,101)(9,102,12,99)(13,104,16,107)(14,108,17,105)(15,106,18,103)(19,110,22,113)(20,114,23,111)(21,112,24,109)(25,116,28,119)(26,120,29,117)(27,118,30,115)(31,66,35,63)(32,64,36,61)(33,62,34,65)(37,70,41,67)(38,68,42,71)(39,72,40,69)(43,76,47,73)(44,74,48,77)(45,78,46,75)(49,82,53,79)(50,80,54,83)(51,84,52,81)(55,86,58,89)(56,90,59,87)(57,88,60,85), (1,61)(2,65)(3,63)(4,66)(5,64)(6,62)(7,67)(8,71)(9,69)(10,70)(11,68)(12,72)(13,77)(14,75)(15,73)(16,74)(17,78)(18,76)(19,83)(20,81)(21,79)(22,80)(23,84)(24,82)(25,89)(26,87)(27,85)(28,86)(29,90)(30,88)(31,91)(32,95)(33,93)(34,96)(35,94)(36,92)(37,97)(38,101)(39,99)(40,102)(41,100)(42,98)(43,103)(44,107)(45,105)(46,108)(47,106)(48,104)(49,109)(50,113)(51,111)(52,114)(53,112)(54,110)(55,119)(56,117)(57,115)(58,116)(59,120)(60,118), (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), (1,3)(4,5)(7,8)(10,11)(13,15)(16,18)(19,21)(22,24)(25,27)(28,30)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40)(43,48)(44,47)(45,46)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)(61,66)(62,65)(63,64)(67,68)(69,72)(70,71)(73,74)(75,78)(76,77)(79,80)(81,84)(82,83)(85,86)(87,90)(88,89)(91,95)(92,94)(97,101)(98,100)(103,107)(104,106)(109,113)(110,112)(115,119)(116,118) );
G=PermutationGroup([(1,25,19,13,8),(2,26,20,14,9),(3,27,21,15,7),(4,30,24,18,10),(5,28,22,16,11),(6,29,23,17,12),(31,57,49,43,37),(32,55,50,44,38),(33,56,51,45,39),(34,59,52,46,40),(35,60,53,47,41),(36,58,54,48,42),(61,89,83,77,71),(62,90,84,78,72),(63,85,79,73,67),(64,86,80,74,68),(65,87,81,75,69),(66,88,82,76,70),(91,115,109,103,97),(92,116,110,104,98),(93,117,111,105,99),(94,118,112,106,100),(95,119,113,107,101),(96,120,114,108,102)], [(1,92,5,95),(2,96,6,93),(3,94,4,91),(7,100,10,97),(8,98,11,101),(9,102,12,99),(13,104,16,107),(14,108,17,105),(15,106,18,103),(19,110,22,113),(20,114,23,111),(21,112,24,109),(25,116,28,119),(26,120,29,117),(27,118,30,115),(31,66,35,63),(32,64,36,61),(33,62,34,65),(37,70,41,67),(38,68,42,71),(39,72,40,69),(43,76,47,73),(44,74,48,77),(45,78,46,75),(49,82,53,79),(50,80,54,83),(51,84,52,81),(55,86,58,89),(56,90,59,87),(57,88,60,85)], [(1,61),(2,65),(3,63),(4,66),(5,64),(6,62),(7,67),(8,71),(9,69),(10,70),(11,68),(12,72),(13,77),(14,75),(15,73),(16,74),(17,78),(18,76),(19,83),(20,81),(21,79),(22,80),(23,84),(24,82),(25,89),(26,87),(27,85),(28,86),(29,90),(30,88),(31,91),(32,95),(33,93),(34,96),(35,94),(36,92),(37,97),(38,101),(39,99),(40,102),(41,100),(42,98),(43,103),(44,107),(45,105),(46,108),(47,106),(48,104),(49,109),(50,113),(51,111),(52,114),(53,112),(54,110),(55,119),(56,117),(57,115),(58,116),(59,120),(60,118)], [(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)], [(1,3),(4,5),(7,8),(10,11),(13,15),(16,18),(19,21),(22,24),(25,27),(28,30),(31,36),(32,35),(33,34),(37,42),(38,41),(39,40),(43,48),(44,47),(45,46),(49,54),(50,53),(51,52),(55,60),(56,59),(57,58),(61,66),(62,65),(63,64),(67,68),(69,72),(70,71),(73,74),(75,78),(76,77),(79,80),(81,84),(82,83),(85,86),(87,90),(88,89),(91,95),(92,94),(97,101),(98,100),(103,107),(104,106),(109,113),(110,112),(115,119),(116,118)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 52 | 43 | 18 | 36 |
| 18 | 9 | 25 | 43 |
| 52 | 43 | 9 | 18 |
| 18 | 9 | 43 | 52 |
| 1 | 0 | 59 | 0 |
| 0 | 1 | 0 | 59 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 60 |
| 60 | 60 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 60 |
| 1 | 0 | 60 | 0 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[52,18,52,18,43,9,43,9,18,25,9,43,36,43,18,52],[1,0,0,0,0,1,0,0,59,0,60,0,0,59,0,60],[0,60,0,60,1,60,1,60,0,0,0,1,0,0,60,1],[0,1,0,1,1,0,1,0,0,0,0,60,0,0,60,0] >;
135 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 10D | 10E | ··· | 10X | 10Y | ··· | 10AN | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
135 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 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D6 | D6 | D6 | C5×S3 | S3×C10 | S3×C10 | S3×C10 | 2+ (1+4) | D4⋊6D6 | C5×2+ (1+4) | C5×D4⋊6D6 |
| kernel | C5×D4⋊6D6 | C5×C4○D12 | C5×S3×D4 | C5×D4⋊2S3 | C10×C3⋊D4 | D4×C30 | D4⋊6D6 | C4○D12 | S3×D4 | D4⋊2S3 | C2×C3⋊D4 | C6×D4 | D4×C10 | C2×C20 | C5×D4 | C22×C10 | C2×D4 | C2×C4 | D4 | C23 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 4 | 8 | 16 | 16 | 16 | 4 | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_6D_6
% in TeX
G:=Group("C5xD4:6D6"); // GroupNames label
G:=SmallGroup(480,1156);
// by ID
G=gap.SmallGroup(480,1156);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations