direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D12⋊6C22, C60.148D4, C60.228C23, D4⋊S3⋊5C10, (C6×D4)⋊2C10, (D4×C10)⋊9S3, C4○D12⋊3C10, (D4×C30)⋊16C2, D12⋊6(C2×C10), D4.S3⋊5C10, (C5×D4).35D6, D4.6(S3×C10), C12.15(C5×D4), C6.45(D4×C10), C15⋊36(C8⋊C22), Dic6⋊5(C2×C10), (C2×C30).182D4, C30.428(C2×D4), (C2×C20).242D6, C4.Dic3⋊6C10, (C5×D12)⋊36C22, C20.95(C3⋊D4), C20.201(C22×S3), (C2×C60).358C22, C12.12(C22×C10), (C5×Dic6)⋊32C22, (D4×C15).45C22, C3⋊C8⋊3(C2×C10), C3⋊4(C5×C8⋊C22), (C2×D4)⋊2(C5×S3), C4.12(S3×C2×C10), (C5×D4⋊S3)⋊13C2, (C5×C3⋊C8)⋊25C22, (C2×C6).39(C5×D4), C4.16(C5×C3⋊D4), C2.9(C10×C3⋊D4), (C5×C4○D12)⋊13C2, (C2×C4).15(S3×C10), (C5×D4.S3)⋊13C2, (C3×D4).6(C2×C10), (C2×C12).31(C2×C10), C10.130(C2×C3⋊D4), (C5×C4.Dic3)⋊18C2, C22.10(C5×C3⋊D4), (C2×C10).63(C3⋊D4), SmallGroup(480,811)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D12⋊6C22
G = < a,b,c,d,e | a5=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >
Subgroups: 324 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, C5×S3, C30, C30, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, D12⋊6C22, C5×C3⋊C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4×C15, D4×C15, C22×C30, C5×C8⋊C22, C5×C4.Dic3, C5×D4⋊S3, C5×D4.S3, C5×C4○D12, D4×C30, C5×D12⋊6C22
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, C2×C3⋊D4, S3×C10, D4×C10, D12⋊6C22, C5×C3⋊D4, S3×C2×C10, C5×C8⋊C22, C10×C3⋊D4, C5×D12⋊6C22
►On 120 points
Generators in S120
(1 56 45 35 18)(2 57 46 36 19)(3 58 47 25 20)(4 59 48 26 21)(5 60 37 27 22)(6 49 38 28 23)(7 50 39 29 24)(8 51 40 30 13)(9 52 41 31 14)(10 53 42 32 15)(11 54 43 33 16)(12 55 44 34 17)(61 118 106 94 73)(62 119 107 95 74)(63 120 108 96 75)(64 109 97 85 76)(65 110 98 86 77)(66 111 99 87 78)(67 112 100 88 79)(68 113 101 89 80)(69 114 102 90 81)(70 115 103 91 82)(71 116 104 92 83)(72 117 105 93 84)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 74)(14 73)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 98)(38 97)(39 108)(40 107)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 100)(48 99)(49 109)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)
(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(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)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(1 4)(2 11)(3 6)(5 8)(7 10)(9 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)(25 28)(26 35)(27 30)(29 32)(31 34)(33 36)(37 40)(38 47)(39 42)(41 44)(43 46)(45 48)(49 58)(50 53)(51 60)(52 55)(54 57)(56 59)(61 67)(63 69)(65 71)(73 79)(75 81)(77 83)(86 92)(88 94)(90 96)(98 104)(100 106)(102 108)(110 116)(112 118)(114 120)
G:=sub<Sym(120)| (1,56,45,35,18)(2,57,46,36,19)(3,58,47,25,20)(4,59,48,26,21)(5,60,37,27,22)(6,49,38,28,23)(7,50,39,29,24)(8,51,40,30,13)(9,52,41,31,14)(10,53,42,32,15)(11,54,43,33,16)(12,55,44,34,17)(61,118,106,94,73)(62,119,107,95,74)(63,120,108,96,75)(64,109,97,85,76)(65,110,98,86,77)(66,111,99,87,78)(67,112,100,88,79)(68,113,101,89,80)(69,114,102,90,81)(70,115,103,91,82)(71,116,104,92,83)(72,117,105,93,84), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,72)(11,71)(12,70)(13,74)(14,73)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,98)(38,97)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,109)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110), (61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(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)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120), (1,4)(2,11)(3,6)(5,8)(7,10)(9,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,40)(38,47)(39,42)(41,44)(43,46)(45,48)(49,58)(50,53)(51,60)(52,55)(54,57)(56,59)(61,67)(63,69)(65,71)(73,79)(75,81)(77,83)(86,92)(88,94)(90,96)(98,104)(100,106)(102,108)(110,116)(112,118)(114,120)>;
G:=Group( (1,56,45,35,18)(2,57,46,36,19)(3,58,47,25,20)(4,59,48,26,21)(5,60,37,27,22)(6,49,38,28,23)(7,50,39,29,24)(8,51,40,30,13)(9,52,41,31,14)(10,53,42,32,15)(11,54,43,33,16)(12,55,44,34,17)(61,118,106,94,73)(62,119,107,95,74)(63,120,108,96,75)(64,109,97,85,76)(65,110,98,86,77)(66,111,99,87,78)(67,112,100,88,79)(68,113,101,89,80)(69,114,102,90,81)(70,115,103,91,82)(71,116,104,92,83)(72,117,105,93,84), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,72)(11,71)(12,70)(13,74)(14,73)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,98)(38,97)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,109)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110), (61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(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)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120), (1,4)(2,11)(3,6)(5,8)(7,10)(9,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,40)(38,47)(39,42)(41,44)(43,46)(45,48)(49,58)(50,53)(51,60)(52,55)(54,57)(56,59)(61,67)(63,69)(65,71)(73,79)(75,81)(77,83)(86,92)(88,94)(90,96)(98,104)(100,106)(102,108)(110,116)(112,118)(114,120) );
G=PermutationGroup([[(1,56,45,35,18),(2,57,46,36,19),(3,58,47,25,20),(4,59,48,26,21),(5,60,37,27,22),(6,49,38,28,23),(7,50,39,29,24),(8,51,40,30,13),(9,52,41,31,14),(10,53,42,32,15),(11,54,43,33,16),(12,55,44,34,17),(61,118,106,94,73),(62,119,107,95,74),(63,120,108,96,75),(64,109,97,85,76),(65,110,98,86,77),(66,111,99,87,78),(67,112,100,88,79),(68,113,101,89,80),(69,114,102,90,81),(70,115,103,91,82),(71,116,104,92,83),(72,117,105,93,84)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,72),(11,71),(12,70),(13,74),(14,73),(15,84),(16,83),(17,82),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,88),(26,87),(27,86),(28,85),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,98),(38,97),(39,108),(40,107),(41,106),(42,105),(43,104),(44,103),(45,102),(46,101),(47,100),(48,99),(49,109),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110)], [(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(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),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120)], [(1,4),(2,11),(3,6),(5,8),(7,10),(9,12),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23),(25,28),(26,35),(27,30),(29,32),(31,34),(33,36),(37,40),(38,47),(39,42),(41,44),(43,46),(45,48),(49,58),(50,53),(51,60),(52,55),(54,57),(56,59),(61,67),(63,69),(65,71),(73,79),(75,81),(77,83),(86,92),(88,94),(90,96),(98,104),(100,106),(102,108),(110,116),(112,118),(114,120)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 10Q | 10R | 10S | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 30A | ··· | 30L | 30M | ··· | 30AB | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 4 | 12 | 2 | 2 | 2 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
105 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 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×C3⋊D4 | C5×C3⋊D4 | C8⋊C22 | D12⋊6C22 | C5×C8⋊C22 | C5×D12⋊6C22 |
| kernel | C5×D12⋊6C22 | C5×C4.Dic3 | C5×D4⋊S3 | C5×D4.S3 | C5×C4○D12 | D4×C30 | D12⋊6C22 | C4.Dic3 | D4⋊S3 | D4.S3 | C4○D12 | C6×D4 | D4×C10 | C60 | C2×C30 | C2×C20 | C5×D4 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×D12⋊6C22 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 225 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 226 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 226 |
| 0 | 0 | 15 | 0 |
| 0 | 225 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[0,16,0,0,225,0,0,0,0,0,0,15,0,0,226,0],[0,0,0,16,0,0,225,0,0,15,0,0,226,0,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,240] >;
C5×D12⋊6C22 in GAP, Magma, Sage, TeX
C_5\times D_{12}\rtimes_6C_2^2 % in TeX
G:=Group("C5xD12:6C2^2"); // GroupNames label
G:=SmallGroup(480,811);
// by ID
G=gap.SmallGroup(480,811);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,4204,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^12=c^2=d^2=e^2=1,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,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations