direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C6.D4, C30.51D4, (C2×C6)⋊2C20, (C2×C30)⋊10C4, C6.9(C2×C20), C6.11(C5×D4), C30.61(C2×C4), (C2×C10)⋊6Dic3, (C2×C10).35D6, C23.2(C5×S3), C15⋊12(C22⋊C4), (C10×Dic3)⋊8C2, (C2×Dic3)⋊2C10, (C22×C10).3S3, (C22×C30).6C2, C22.7(S3×C10), (C22×C6).2C10, C2.5(C10×Dic3), C22⋊2(C5×Dic3), C10.27(C3⋊D4), (C2×C30).46C22, C10.22(C2×Dic3), C3⋊2(C5×C22⋊C4), C2.3(C5×C3⋊D4), (C2×C6).7(C2×C10), SmallGroup(240,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C6.D4
G = < a,b,c,d | a5=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 120 in 68 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C10, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C22×C6, C30, C30, C30, C2×C20, C22×C10, C6.D4, C5×Dic3, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C10×Dic3, C22×C30, C5×C6.D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C10×Dic3, C5×C3⋊D4, C5×C6.D4
►On 120 points
Generators in S120
(1 66 49 37 25)(2 61 50 38 26)(3 62 51 39 27)(4 63 52 40 28)(5 64 53 41 29)(6 65 54 42 30)(7 110 98 86 74)(8 111 99 87 75)(9 112 100 88 76)(10 113 101 89 77)(11 114 102 90 78)(12 109 97 85 73)(13 58 46 34 22)(14 59 47 35 23)(15 60 48 36 24)(16 55 43 31 19)(17 56 44 32 20)(18 57 45 33 21)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 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 67 14 74)(2 72 15 73)(3 71 16 78)(4 70 17 77)(5 69 18 76)(6 68 13 75)(7 66 115 59)(8 65 116 58)(9 64 117 57)(10 63 118 56)(11 62 119 55)(12 61 120 60)(19 90 27 83)(20 89 28 82)(21 88 29 81)(22 87 30 80)(23 86 25 79)(24 85 26 84)(31 102 39 95)(32 101 40 94)(33 100 41 93)(34 99 42 92)(35 98 37 91)(36 97 38 96)(43 114 51 107)(44 113 52 106)(45 112 53 105)(46 111 54 104)(47 110 49 103)(48 109 50 108)
(1 77 4 74)(2 76 5 73)(3 75 6 78)(7 66 10 63)(8 65 11 62)(9 64 12 61)(13 71 16 68)(14 70 17 67)(15 69 18 72)(19 80 22 83)(20 79 23 82)(21 84 24 81)(25 89 28 86)(26 88 29 85)(27 87 30 90)(31 92 34 95)(32 91 35 94)(33 96 36 93)(37 101 40 98)(38 100 41 97)(39 99 42 102)(43 104 46 107)(44 103 47 106)(45 108 48 105)(49 113 52 110)(50 112 53 109)(51 111 54 114)(55 116 58 119)(56 115 59 118)(57 120 60 117)
G:=sub<Sym(120)| (1,66,49,37,25)(2,61,50,38,26)(3,62,51,39,27)(4,63,52,40,28)(5,64,53,41,29)(6,65,54,42,30)(7,110,98,86,74)(8,111,99,87,75)(9,112,100,88,76)(10,113,101,89,77)(11,114,102,90,78)(12,109,97,85,73)(13,58,46,34,22)(14,59,47,35,23)(15,60,48,36,24)(16,55,43,31,19)(17,56,44,32,20)(18,57,45,33,21)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84), (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), (1,67,14,74)(2,72,15,73)(3,71,16,78)(4,70,17,77)(5,69,18,76)(6,68,13,75)(7,66,115,59)(8,65,116,58)(9,64,117,57)(10,63,118,56)(11,62,119,55)(12,61,120,60)(19,90,27,83)(20,89,28,82)(21,88,29,81)(22,87,30,80)(23,86,25,79)(24,85,26,84)(31,102,39,95)(32,101,40,94)(33,100,41,93)(34,99,42,92)(35,98,37,91)(36,97,38,96)(43,114,51,107)(44,113,52,106)(45,112,53,105)(46,111,54,104)(47,110,49,103)(48,109,50,108), (1,77,4,74)(2,76,5,73)(3,75,6,78)(7,66,10,63)(8,65,11,62)(9,64,12,61)(13,71,16,68)(14,70,17,67)(15,69,18,72)(19,80,22,83)(20,79,23,82)(21,84,24,81)(25,89,28,86)(26,88,29,85)(27,87,30,90)(31,92,34,95)(32,91,35,94)(33,96,36,93)(37,101,40,98)(38,100,41,97)(39,99,42,102)(43,104,46,107)(44,103,47,106)(45,108,48,105)(49,113,52,110)(50,112,53,109)(51,111,54,114)(55,116,58,119)(56,115,59,118)(57,120,60,117)>;
G:=Group( (1,66,49,37,25)(2,61,50,38,26)(3,62,51,39,27)(4,63,52,40,28)(5,64,53,41,29)(6,65,54,42,30)(7,110,98,86,74)(8,111,99,87,75)(9,112,100,88,76)(10,113,101,89,77)(11,114,102,90,78)(12,109,97,85,73)(13,58,46,34,22)(14,59,47,35,23)(15,60,48,36,24)(16,55,43,31,19)(17,56,44,32,20)(18,57,45,33,21)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84), (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), (1,67,14,74)(2,72,15,73)(3,71,16,78)(4,70,17,77)(5,69,18,76)(6,68,13,75)(7,66,115,59)(8,65,116,58)(9,64,117,57)(10,63,118,56)(11,62,119,55)(12,61,120,60)(19,90,27,83)(20,89,28,82)(21,88,29,81)(22,87,30,80)(23,86,25,79)(24,85,26,84)(31,102,39,95)(32,101,40,94)(33,100,41,93)(34,99,42,92)(35,98,37,91)(36,97,38,96)(43,114,51,107)(44,113,52,106)(45,112,53,105)(46,111,54,104)(47,110,49,103)(48,109,50,108), (1,77,4,74)(2,76,5,73)(3,75,6,78)(7,66,10,63)(8,65,11,62)(9,64,12,61)(13,71,16,68)(14,70,17,67)(15,69,18,72)(19,80,22,83)(20,79,23,82)(21,84,24,81)(25,89,28,86)(26,88,29,85)(27,87,30,90)(31,92,34,95)(32,91,35,94)(33,96,36,93)(37,101,40,98)(38,100,41,97)(39,99,42,102)(43,104,46,107)(44,103,47,106)(45,108,48,105)(49,113,52,110)(50,112,53,109)(51,111,54,114)(55,116,58,119)(56,115,59,118)(57,120,60,117) );
G=PermutationGroup([[(1,66,49,37,25),(2,61,50,38,26),(3,62,51,39,27),(4,63,52,40,28),(5,64,53,41,29),(6,65,54,42,30),(7,110,98,86,74),(8,111,99,87,75),(9,112,100,88,76),(10,113,101,89,77),(11,114,102,90,78),(12,109,97,85,73),(13,58,46,34,22),(14,59,47,35,23),(15,60,48,36,24),(16,55,43,31,19),(17,56,44,32,20),(18,57,45,33,21),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84)], [(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)], [(1,67,14,74),(2,72,15,73),(3,71,16,78),(4,70,17,77),(5,69,18,76),(6,68,13,75),(7,66,115,59),(8,65,116,58),(9,64,117,57),(10,63,118,56),(11,62,119,55),(12,61,120,60),(19,90,27,83),(20,89,28,82),(21,88,29,81),(22,87,30,80),(23,86,25,79),(24,85,26,84),(31,102,39,95),(32,101,40,94),(33,100,41,93),(34,99,42,92),(35,98,37,91),(36,97,38,96),(43,114,51,107),(44,113,52,106),(45,112,53,105),(46,111,54,104),(47,110,49,103),(48,109,50,108)], [(1,77,4,74),(2,76,5,73),(3,75,6,78),(7,66,10,63),(8,65,11,62),(9,64,12,61),(13,71,16,68),(14,70,17,67),(15,69,18,72),(19,80,22,83),(20,79,23,82),(21,84,24,81),(25,89,28,86),(26,88,29,85),(27,87,30,90),(31,92,34,95),(32,91,35,94),(33,96,36,93),(37,101,40,98),(38,100,41,97),(39,99,42,102),(43,104,46,107),(44,103,47,106),(45,108,48,105),(49,113,52,110),(50,112,53,109),(51,111,54,114),(55,116,58,119),(56,115,59,118),(57,120,60,117)]])
C5×C6.D4 is a maximal subgroup of
(C2×C6).D20 C15⋊9(C23⋊C4) Dic15.19D4 (C6×Dic5)⋊7C4 C23.13(S3×D5) C23.14(S3×D5) C23.48(S3×D5) D30⋊6D4 C6.(D4×D5) C6.(C2×D20) C6.D4⋊D5 C23.17(S3×D5) Dic15⋊3D4 C15⋊26(C4×D4) Dic15⋊16D4 D30.45D4 D30.16D4 (C2×C6)⋊8D20 D30⋊18D4 (C2×C30)⋊Q8 Dic15.48D4 C5×S3×C22⋊C4 C20×C3⋊D4 C5×D4×Dic3
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | D4 | Dic3 | D6 | C3⋊D4 | C5×S3 | C5×D4 | C5×Dic3 | S3×C10 | C5×C3⋊D4 |
| kernel | C5×C6.D4 | C10×Dic3 | C22×C30 | C2×C30 | C6.D4 | C2×Dic3 | C22×C6 | C2×C6 | C22×C10 | C30 | C2×C10 | C2×C10 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 2 | 2 | 1 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C5×C6.D4 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 58 | 0 |
| 0 | 0 | 58 |
| 60 | 0 | 0 |
| 0 | 47 | 0 |
| 0 | 0 | 13 |
| 11 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 0 |
| 50 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(61))| [1,0,0,0,58,0,0,0,58],[60,0,0,0,47,0,0,0,13],[11,0,0,0,0,60,0,1,0],[50,0,0,0,0,1,0,1,0] >;
C5×C6.D4 in GAP, Magma, Sage, TeX
C_5\times C_6.D_4
% in TeX
G:=Group("C5xC6.D4"); // GroupNames label
G:=SmallGroup(240,64);
// by ID
G=gap.SmallGroup(240,64);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,505,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations