direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C5⋊D4, C30⋊8D4, C30.47C23, C5⋊3(C6×D4), C10⋊2(C3×D4), (C2×C6)⋊7D10, C15⋊17(C2×D4), D10⋊3(C2×C6), C23⋊2(C3×D5), (C22×C6)⋊1D5, C22⋊3(C6×D5), (C22×C10)⋊4C6, (C22×C30)⋊4C2, Dic5⋊2(C2×C6), (C2×Dic5)⋊4C6, (C22×D5)⋊4C6, (C2×C30)⋊11C22, (C6×Dic5)⋊10C2, (C6×D5)⋊11C22, C6.47(C22×D5), C10.10(C22×C6), (C3×Dic5)⋊9C22, (D5×C2×C6)⋊7C2, C2.10(D5×C2×C6), (C2×C10)⋊5(C2×C6), SmallGroup(240,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C5⋊D4
G = < a,b,c,d | a6=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 292 in 108 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C2×D4, Dic5, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C6×D4, C3×Dic5, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C2×C5⋊D4, C6×Dic5, C3×C5⋊D4, D5×C2×C6, C22×C30, C6×C5⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C5⋊D4, D5×C2×C6, C6×C5⋊D4
►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 17 109 50 115)(2 18 110 51 116)(3 13 111 52 117)(4 14 112 53 118)(5 15 113 54 119)(6 16 114 49 120)(7 103 19 46 92)(8 104 20 47 93)(9 105 21 48 94)(10 106 22 43 95)(11 107 23 44 96)(12 108 24 45 91)(25 55 67 31 79)(26 56 68 32 80)(27 57 69 33 81)(28 58 70 34 82)(29 59 71 35 83)(30 60 72 36 84)(37 77 89 101 65)(38 78 90 102 66)(39 73 85 97 61)(40 74 86 98 62)(41 75 87 99 63)(42 76 88 100 64)
(1 89 30 95)(2 90 25 96)(3 85 26 91)(4 86 27 92)(5 87 28 93)(6 88 29 94)(7 118 98 81)(8 119 99 82)(9 120 100 83)(10 115 101 84)(11 116 102 79)(12 117 97 80)(13 73 56 45)(14 74 57 46)(15 75 58 47)(16 76 59 48)(17 77 60 43)(18 78 55 44)(19 112 40 69)(20 113 41 70)(21 114 42 71)(22 109 37 72)(23 110 38 67)(24 111 39 68)(31 107 51 66)(32 108 52 61)(33 103 53 62)(34 104 54 63)(35 105 49 64)(36 106 50 65)
(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 117)(14 118)(15 119)(16 120)(17 115)(18 116)(19 62)(20 63)(21 64)(22 65)(23 66)(24 61)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 106)(38 107)(39 108)(40 103)(41 104)(42 105)(43 101)(44 102)(45 97)(46 98)(47 99)(48 100)(49 114)(50 109)(51 110)(52 111)(53 112)(54 113)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
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), (1,17,109,50,115)(2,18,110,51,116)(3,13,111,52,117)(4,14,112,53,118)(5,15,113,54,119)(6,16,114,49,120)(7,103,19,46,92)(8,104,20,47,93)(9,105,21,48,94)(10,106,22,43,95)(11,107,23,44,96)(12,108,24,45,91)(25,55,67,31,79)(26,56,68,32,80)(27,57,69,33,81)(28,58,70,34,82)(29,59,71,35,83)(30,60,72,36,84)(37,77,89,101,65)(38,78,90,102,66)(39,73,85,97,61)(40,74,86,98,62)(41,75,87,99,63)(42,76,88,100,64), (1,89,30,95)(2,90,25,96)(3,85,26,91)(4,86,27,92)(5,87,28,93)(6,88,29,94)(7,118,98,81)(8,119,99,82)(9,120,100,83)(10,115,101,84)(11,116,102,79)(12,117,97,80)(13,73,56,45)(14,74,57,46)(15,75,58,47)(16,76,59,48)(17,77,60,43)(18,78,55,44)(19,112,40,69)(20,113,41,70)(21,114,42,71)(22,109,37,72)(23,110,38,67)(24,111,39,68)(31,107,51,66)(32,108,52,61)(33,103,53,62)(34,104,54,63)(35,105,49,64)(36,106,50,65), (7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,117)(14,118)(15,119)(16,120)(17,115)(18,116)(19,62)(20,63)(21,64)(22,65)(23,66)(24,61)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,106)(38,107)(39,108)(40,103)(41,104)(42,105)(43,101)(44,102)(45,97)(46,98)(47,99)(48,100)(49,114)(50,109)(51,110)(52,111)(53,112)(54,113)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)>;
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), (1,17,109,50,115)(2,18,110,51,116)(3,13,111,52,117)(4,14,112,53,118)(5,15,113,54,119)(6,16,114,49,120)(7,103,19,46,92)(8,104,20,47,93)(9,105,21,48,94)(10,106,22,43,95)(11,107,23,44,96)(12,108,24,45,91)(25,55,67,31,79)(26,56,68,32,80)(27,57,69,33,81)(28,58,70,34,82)(29,59,71,35,83)(30,60,72,36,84)(37,77,89,101,65)(38,78,90,102,66)(39,73,85,97,61)(40,74,86,98,62)(41,75,87,99,63)(42,76,88,100,64), (1,89,30,95)(2,90,25,96)(3,85,26,91)(4,86,27,92)(5,87,28,93)(6,88,29,94)(7,118,98,81)(8,119,99,82)(9,120,100,83)(10,115,101,84)(11,116,102,79)(12,117,97,80)(13,73,56,45)(14,74,57,46)(15,75,58,47)(16,76,59,48)(17,77,60,43)(18,78,55,44)(19,112,40,69)(20,113,41,70)(21,114,42,71)(22,109,37,72)(23,110,38,67)(24,111,39,68)(31,107,51,66)(32,108,52,61)(33,103,53,62)(34,104,54,63)(35,105,49,64)(36,106,50,65), (7,74)(8,75)(9,76)(10,77)(11,78)(12,73)(13,117)(14,118)(15,119)(16,120)(17,115)(18,116)(19,62)(20,63)(21,64)(22,65)(23,66)(24,61)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,106)(38,107)(39,108)(40,103)(41,104)(42,105)(43,101)(44,102)(45,97)(46,98)(47,99)(48,100)(49,114)(50,109)(51,110)(52,111)(53,112)(54,113)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96) );
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)], [(1,17,109,50,115),(2,18,110,51,116),(3,13,111,52,117),(4,14,112,53,118),(5,15,113,54,119),(6,16,114,49,120),(7,103,19,46,92),(8,104,20,47,93),(9,105,21,48,94),(10,106,22,43,95),(11,107,23,44,96),(12,108,24,45,91),(25,55,67,31,79),(26,56,68,32,80),(27,57,69,33,81),(28,58,70,34,82),(29,59,71,35,83),(30,60,72,36,84),(37,77,89,101,65),(38,78,90,102,66),(39,73,85,97,61),(40,74,86,98,62),(41,75,87,99,63),(42,76,88,100,64)], [(1,89,30,95),(2,90,25,96),(3,85,26,91),(4,86,27,92),(5,87,28,93),(6,88,29,94),(7,118,98,81),(8,119,99,82),(9,120,100,83),(10,115,101,84),(11,116,102,79),(12,117,97,80),(13,73,56,45),(14,74,57,46),(15,75,58,47),(16,76,59,48),(17,77,60,43),(18,78,55,44),(19,112,40,69),(20,113,41,70),(21,114,42,71),(22,109,37,72),(23,110,38,67),(24,111,39,68),(31,107,51,66),(32,108,52,61),(33,103,53,62),(34,104,54,63),(35,105,49,64),(36,106,50,65)], [(7,74),(8,75),(9,76),(10,77),(11,78),(12,73),(13,117),(14,118),(15,119),(16,120),(17,115),(18,116),(19,62),(20,63),(21,64),(22,65),(23,66),(24,61),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,106),(38,107),(39,108),(40,103),(41,104),(42,105),(43,101),(44,102),(45,97),(46,98),(47,99),(48,100),(49,114),(50,109),(51,110),(52,111),(53,112),(54,113),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)]])
C6×C5⋊D4 is a maximal subgroup of
(C2×C6).D20 C3⋊(C23⋊F5) C5⋊(C12.D4) D30⋊6D4 C6.(D4×D5) (C2×C30).D4 C6.(C2×D20) C23.17(S3×D5) (C6×D5)⋊D4 Dic15⋊3D4 Dic15⋊16D4 (S3×C10)⋊D4 Dic15⋊5D4 C15⋊C22≀C2 (C2×C6)⋊D20 Dic15⋊18D4 D30⋊18D4 D30⋊8D4 C15⋊2+ 1+4 C6×D4×D5
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | 1 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D5 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | C3×C5⋊D4 |
| kernel | C6×C5⋊D4 | C6×Dic5 | C3×C5⋊D4 | D5×C2×C6 | C22×C30 | C2×C5⋊D4 | C2×Dic5 | C5⋊D4 | C22×D5 | C22×C10 | C30 | C22×C6 | C2×C6 | C10 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 8 | 12 | 16 |
Matrix representation of C6×C5⋊D4 ►in GL3(𝔽61) generated by
| 48 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 1 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 1 | 43 |
| 60 | 0 | 0 |
| 0 | 31 | 8 |
| 0 | 17 | 30 |
| 60 | 0 | 0 |
| 0 | 1 | 43 |
| 0 | 0 | 60 |
G:=sub<GL(3,GF(61))| [48,0,0,0,60,0,0,0,60],[1,0,0,0,0,1,0,60,43],[60,0,0,0,31,17,0,8,30],[60,0,0,0,1,0,0,43,60] >;
C6×C5⋊D4 in GAP, Magma, Sage, TeX
C_6\times C_5\rtimes D_4
% in TeX
G:=Group("C6xC5:D4"); // GroupNames label
G:=SmallGroup(240,164);
// by ID
G=gap.SmallGroup(240,164);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,506,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations