direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4○D12, C30.95C24, C60.242C23, C15⋊142+ 1+4, (C5×D4)⋊30D6, (S3×D4)⋊5C10, D4⋊8(S3×C10), (C2×C20)⋊23D6, Q8⋊8(S3×C10), (C5×Q8)⋊29D6, C4○D12⋊8C10, (C2×D12)⋊13C10, D12⋊11(C2×C10), (C10×D12)⋊29C2, (C2×C60)⋊30C22, Q8⋊3S3⋊5C10, (S3×C20)⋊15C22, Dic6⋊12(C2×C10), C3⋊2(C5×2+ 1+4), (C5×D12)⋊41C22, (D4×C15)⋊40C22, C6.12(C23×C10), C10.80(S3×C23), (Q8×C15)⋊35C22, D6.6(C22×C10), (S3×C10).42C23, C20.239(C22×S3), (C2×C30).260C23, C12.26(C22×C10), (C5×Dic6)⋊39C22, Dic3.8(C22×C10), (C5×Dic3).44C23, (C5×S3×D4)⋊12C2, (C2×C4)⋊4(S3×C10), C4○D4⋊5(C5×S3), C4.26(S3×C2×C10), (C4×S3)⋊2(C2×C10), (C2×C12)⋊5(C2×C10), (C3×C4○D4)⋊4C10, (C5×C4○D4)⋊12S3, (C3×D4)⋊9(C2×C10), C3⋊D4⋊5(C2×C10), (C3×Q8)⋊8(C2×C10), C22.4(S3×C2×C10), (C5×C4○D12)⋊18C2, (C15×C4○D4)⋊14C2, (S3×C2×C10)⋊16C22, C2.13(S3×C22×C10), (C22×S3)⋊4(C2×C10), (C5×Q8⋊3S3)⋊12C2, (C5×C3⋊D4)⋊21C22, (C2×C6).4(C22×C10), (C2×C10).23(C22×S3), SmallGroup(480,1161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4○D12
G = < a,b,c,d,e | a5=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >
Subgroups: 788 in 332 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×D4, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C5×S3, C30, C30, 2+ 1+4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, D4×C10, C5×C4○D4, C5×C4○D4, D4○D12, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4×C15, Q8×C15, S3×C2×C10, C5×2+ 1+4, C10×D12, C5×C4○D12, C5×S3×D4, C5×Q8⋊3S3, C15×C4○D4, C5×D4○D12
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C24, C2×C10, C22×S3, C5×S3, 2+ 1+4, C22×C10, S3×C23, S3×C10, C23×C10, D4○D12, S3×C2×C10, C5×2+ 1+4, S3×C22×C10, C5×D4○D12
►On 120 points
Generators in S120
(1 70 21 114 33)(2 71 22 115 34)(3 72 23 116 35)(4 61 24 117 36)(5 62 13 118 25)(6 63 14 119 26)(7 64 15 120 27)(8 65 16 109 28)(9 66 17 110 29)(10 67 18 111 30)(11 68 19 112 31)(12 69 20 113 32)(37 98 58 93 83)(38 99 59 94 84)(39 100 60 95 73)(40 101 49 96 74)(41 102 50 85 75)(42 103 51 86 76)(43 104 52 87 77)(44 105 53 88 78)(45 106 54 89 79)(46 107 55 90 80)(47 108 56 91 81)(48 97 57 92 82)
(1 60 7 54)(2 49 8 55)(3 50 9 56)(4 51 10 57)(5 52 11 58)(6 53 12 59)(13 77 19 83)(14 78 20 84)(15 79 21 73)(16 80 22 74)(17 81 23 75)(18 82 24 76)(25 104 31 98)(26 105 32 99)(27 106 33 100)(28 107 34 101)(29 108 35 102)(30 97 36 103)(37 118 43 112)(38 119 44 113)(39 120 45 114)(40 109 46 115)(41 110 47 116)(42 111 48 117)(61 86 67 92)(62 87 68 93)(63 88 69 94)(64 89 70 95)(65 90 71 96)(66 91 72 85)
(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(97 102)(98 101)(99 100)(103 108)(104 107)(105 106)(109 118)(110 117)(111 116)(112 115)(113 114)(119 120)
G:=sub<Sym(120)| (1,70,21,114,33)(2,71,22,115,34)(3,72,23,116,35)(4,61,24,117,36)(5,62,13,118,25)(6,63,14,119,26)(7,64,15,120,27)(8,65,16,109,28)(9,66,17,110,29)(10,67,18,111,30)(11,68,19,112,31)(12,69,20,113,32)(37,98,58,93,83)(38,99,59,94,84)(39,100,60,95,73)(40,101,49,96,74)(41,102,50,85,75)(42,103,51,86,76)(43,104,52,87,77)(44,105,53,88,78)(45,106,54,89,79)(46,107,55,90,80)(47,108,56,91,81)(48,97,57,92,82), (1,60,7,54)(2,49,8,55)(3,50,9,56)(4,51,10,57)(5,52,11,58)(6,53,12,59)(13,77,19,83)(14,78,20,84)(15,79,21,73)(16,80,22,74)(17,81,23,75)(18,82,24,76)(25,104,31,98)(26,105,32,99)(27,106,33,100)(28,107,34,101)(29,108,35,102)(30,97,36,103)(37,118,43,112)(38,119,44,113)(39,120,45,114)(40,109,46,115)(41,110,47,116)(42,111,48,117)(61,86,67,92)(62,87,68,93)(63,88,69,94)(64,89,70,95)(65,90,71,96)(66,91,72,85), (37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(97,102)(98,101)(99,100)(103,108)(104,107)(105,106)(109,118)(110,117)(111,116)(112,115)(113,114)(119,120)>;
G:=Group( (1,70,21,114,33)(2,71,22,115,34)(3,72,23,116,35)(4,61,24,117,36)(5,62,13,118,25)(6,63,14,119,26)(7,64,15,120,27)(8,65,16,109,28)(9,66,17,110,29)(10,67,18,111,30)(11,68,19,112,31)(12,69,20,113,32)(37,98,58,93,83)(38,99,59,94,84)(39,100,60,95,73)(40,101,49,96,74)(41,102,50,85,75)(42,103,51,86,76)(43,104,52,87,77)(44,105,53,88,78)(45,106,54,89,79)(46,107,55,90,80)(47,108,56,91,81)(48,97,57,92,82), (1,60,7,54)(2,49,8,55)(3,50,9,56)(4,51,10,57)(5,52,11,58)(6,53,12,59)(13,77,19,83)(14,78,20,84)(15,79,21,73)(16,80,22,74)(17,81,23,75)(18,82,24,76)(25,104,31,98)(26,105,32,99)(27,106,33,100)(28,107,34,101)(29,108,35,102)(30,97,36,103)(37,118,43,112)(38,119,44,113)(39,120,45,114)(40,109,46,115)(41,110,47,116)(42,111,48,117)(61,86,67,92)(62,87,68,93)(63,88,69,94)(64,89,70,95)(65,90,71,96)(66,91,72,85), (37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)(61,66)(62,65)(63,64)(67,72)(68,71)(69,70)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95)(97,102)(98,101)(99,100)(103,108)(104,107)(105,106)(109,118)(110,117)(111,116)(112,115)(113,114)(119,120) );
G=PermutationGroup([[(1,70,21,114,33),(2,71,22,115,34),(3,72,23,116,35),(4,61,24,117,36),(5,62,13,118,25),(6,63,14,119,26),(7,64,15,120,27),(8,65,16,109,28),(9,66,17,110,29),(10,67,18,111,30),(11,68,19,112,31),(12,69,20,113,32),(37,98,58,93,83),(38,99,59,94,84),(39,100,60,95,73),(40,101,49,96,74),(41,102,50,85,75),(42,103,51,86,76),(43,104,52,87,77),(44,105,53,88,78),(45,106,54,89,79),(46,107,55,90,80),(47,108,56,91,81),(48,97,57,92,82)], [(1,60,7,54),(2,49,8,55),(3,50,9,56),(4,51,10,57),(5,52,11,58),(6,53,12,59),(13,77,19,83),(14,78,20,84),(15,79,21,73),(16,80,22,74),(17,81,23,75),(18,82,24,76),(25,104,31,98),(26,105,32,99),(27,106,33,100),(28,107,34,101),(29,108,35,102),(30,97,36,103),(37,118,43,112),(38,119,44,113),(39,120,45,114),(40,109,46,115),(41,110,47,116),(42,111,48,117),(61,86,67,92),(62,87,68,93),(63,88,69,94),(64,89,70,95),(65,90,71,96),(66,91,72,85)], [(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60),(61,66),(62,65),(63,64),(67,72),(68,71),(69,70),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,92),(86,91),(87,90),(88,89),(93,96),(94,95),(97,102),(98,101),(99,100),(103,108),(104,107),(105,106),(109,118),(110,117),(111,116),(112,115),(113,114),(119,120)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10AN | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 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 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 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○D12 | C5×2+ 1+4 | C5×D4○D12 |
| kernel | C5×D4○D12 | C10×D12 | C5×C4○D12 | C5×S3×D4 | C5×Q8⋊3S3 | C15×C4○D4 | D4○D12 | C2×D12 | C4○D12 | S3×D4 | Q8⋊3S3 | C3×C4○D4 | C5×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 4 | 12 | 12 | 24 | 8 | 4 | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 1 | 2 | 4 | 8 |
Matrix representation of C5×D4○D12 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 2 |
| 60 | 0 | 60 | 0 |
| 0 | 60 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 60 | 0 |
| 0 | 60 | 0 | 60 |
| 46 | 23 | 0 | 0 |
| 38 | 23 | 0 | 0 |
| 0 | 0 | 46 | 23 |
| 0 | 0 | 38 | 23 |
| 46 | 23 | 0 | 0 |
| 38 | 15 | 0 | 0 |
| 0 | 0 | 46 | 23 |
| 0 | 0 | 38 | 15 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,60,0,0,1,0,60,2,0,60,0,0,2,0,60],[1,0,60,0,0,1,0,60,0,0,60,0,0,0,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[46,38,0,0,23,15,0,0,0,0,46,38,0,0,23,15] >;
C5×D4○D12 in GAP, Magma, Sage, TeX
C_5\times D_4\circ D_{12} % in TeX
G:=Group("C5xD4oD12"); // GroupNames label
G:=SmallGroup(480,1161);
// by ID
G=gap.SmallGroup(480,1161);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,304,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=e^2=1,d^6=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^5>;
// generators/relations