direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.D4, C60.141D4, C12.8(C5×D4), (C6×D4).2C10, (D4×C10).7S3, (D4×C30).12C2, (C2×C20).215D6, C4.Dic3⋊3C10, (C22×C6).2C20, C20.92(C3⋊D4), C15⋊12(C4.D4), (C22×C30).12C4, C23.2(C5×Dic3), (C2×C60).346C22, C22.2(C10×Dic3), (C22×C10).4Dic3, C30.118(C22⋊C4), C10.34(C6.D4), C3⋊2(C5×C4.D4), (C2×C4).3(S3×C10), (C2×D4).2(C5×S3), C4.13(C5×C3⋊D4), (C2×C6).28(C2×C20), C6.14(C5×C22⋊C4), (C2×C30).196(C2×C4), (C2×C12).16(C2×C10), (C5×C4.Dic3)⋊15C2, C2.4(C5×C6.D4), (C2×C10).40(C2×Dic3), SmallGroup(480,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.D4
G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >
Subgroups: 196 in 92 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C3⋊C8, C2×C12, C3×D4, C22×C6, C30, C30, C4.D4, C40, C2×C20, C5×D4, C22×C10, C4.Dic3, C6×D4, C60, C2×C30, C2×C30, C5×M4(2), D4×C10, C12.D4, C5×C3⋊C8, C2×C60, D4×C15, C22×C30, C5×C4.D4, C5×C4.Dic3, D4×C30, C5×C12.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, C4.D4, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C12.D4, C10×Dic3, C5×C3⋊D4, C5×C4.D4, C5×C6.D4, C5×C12.D4
►On 120 points
Generators in S120
(1 55 37 31 13)(2 56 38 32 14)(3 57 39 33 15)(4 58 40 34 16)(5 59 41 35 17)(6 60 42 36 18)(7 49 43 25 19)(8 50 44 26 20)(9 51 45 27 21)(10 52 46 28 22)(11 53 47 29 23)(12 54 48 30 24)(61 112 104 90 77)(62 113 105 91 78)(63 114 106 92 79)(64 115 107 93 80)(65 116 108 94 81)(66 117 97 95 82)(67 118 98 96 83)(68 119 99 85 84)(69 120 100 86 73)(70 109 101 87 74)(71 110 102 88 75)(72 111 103 89 76)
(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 10 72 7 63 4 66)(2 68 11 71 8 62 5 65)(3 67 12 70 9 61 6 64)(13 73 22 76 19 79 16 82)(14 84 23 75 20 78 17 81)(15 83 24 74 21 77 18 80)(25 92 34 95 31 86 28 89)(26 91 35 94 32 85 29 88)(27 90 36 93 33 96 30 87)(37 100 46 103 43 106 40 97)(38 99 47 102 44 105 41 108)(39 98 48 101 45 104 42 107)(49 114 58 117 55 120 52 111)(50 113 59 116 56 119 53 110)(51 112 60 115 57 118 54 109)
(1 63 10 72 7 69 4 66)(2 68 11 65 8 62 5 71)(3 61 12 70 9 67 6 64)(13 79 22 76 19 73 16 82)(14 84 23 81 20 78 17 75)(15 77 24 74 21 83 18 80)(25 86 34 95 31 92 28 89)(26 91 35 88 32 85 29 94)(27 96 36 93 33 90 30 87)(37 106 46 103 43 100 40 97)(38 99 47 108 44 105 41 102)(39 104 48 101 45 98 42 107)(49 120 58 117 55 114 52 111)(50 113 59 110 56 119 53 116)(51 118 60 115 57 112 54 109)
G:=sub<Sym(120)| (1,55,37,31,13)(2,56,38,32,14)(3,57,39,33,15)(4,58,40,34,16)(5,59,41,35,17)(6,60,42,36,18)(7,49,43,25,19)(8,50,44,26,20)(9,51,45,27,21)(10,52,46,28,22)(11,53,47,29,23)(12,54,48,30,24)(61,112,104,90,77)(62,113,105,91,78)(63,114,106,92,79)(64,115,107,93,80)(65,116,108,94,81)(66,117,97,95,82)(67,118,98,96,83)(68,119,99,85,84)(69,120,100,86,73)(70,109,101,87,74)(71,110,102,88,75)(72,111,103,89,76), (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), (1,69,10,72,7,63,4,66)(2,68,11,71,8,62,5,65)(3,67,12,70,9,61,6,64)(13,73,22,76,19,79,16,82)(14,84,23,75,20,78,17,81)(15,83,24,74,21,77,18,80)(25,92,34,95,31,86,28,89)(26,91,35,94,32,85,29,88)(27,90,36,93,33,96,30,87)(37,100,46,103,43,106,40,97)(38,99,47,102,44,105,41,108)(39,98,48,101,45,104,42,107)(49,114,58,117,55,120,52,111)(50,113,59,116,56,119,53,110)(51,112,60,115,57,118,54,109), (1,63,10,72,7,69,4,66)(2,68,11,65,8,62,5,71)(3,61,12,70,9,67,6,64)(13,79,22,76,19,73,16,82)(14,84,23,81,20,78,17,75)(15,77,24,74,21,83,18,80)(25,86,34,95,31,92,28,89)(26,91,35,88,32,85,29,94)(27,96,36,93,33,90,30,87)(37,106,46,103,43,100,40,97)(38,99,47,108,44,105,41,102)(39,104,48,101,45,98,42,107)(49,120,58,117,55,114,52,111)(50,113,59,110,56,119,53,116)(51,118,60,115,57,112,54,109)>;
G:=Group( (1,55,37,31,13)(2,56,38,32,14)(3,57,39,33,15)(4,58,40,34,16)(5,59,41,35,17)(6,60,42,36,18)(7,49,43,25,19)(8,50,44,26,20)(9,51,45,27,21)(10,52,46,28,22)(11,53,47,29,23)(12,54,48,30,24)(61,112,104,90,77)(62,113,105,91,78)(63,114,106,92,79)(64,115,107,93,80)(65,116,108,94,81)(66,117,97,95,82)(67,118,98,96,83)(68,119,99,85,84)(69,120,100,86,73)(70,109,101,87,74)(71,110,102,88,75)(72,111,103,89,76), (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), (1,69,10,72,7,63,4,66)(2,68,11,71,8,62,5,65)(3,67,12,70,9,61,6,64)(13,73,22,76,19,79,16,82)(14,84,23,75,20,78,17,81)(15,83,24,74,21,77,18,80)(25,92,34,95,31,86,28,89)(26,91,35,94,32,85,29,88)(27,90,36,93,33,96,30,87)(37,100,46,103,43,106,40,97)(38,99,47,102,44,105,41,108)(39,98,48,101,45,104,42,107)(49,114,58,117,55,120,52,111)(50,113,59,116,56,119,53,110)(51,112,60,115,57,118,54,109), (1,63,10,72,7,69,4,66)(2,68,11,65,8,62,5,71)(3,61,12,70,9,67,6,64)(13,79,22,76,19,73,16,82)(14,84,23,81,20,78,17,75)(15,77,24,74,21,83,18,80)(25,86,34,95,31,92,28,89)(26,91,35,88,32,85,29,94)(27,96,36,93,33,90,30,87)(37,106,46,103,43,100,40,97)(38,99,47,108,44,105,41,102)(39,104,48,101,45,98,42,107)(49,120,58,117,55,114,52,111)(50,113,59,110,56,119,53,116)(51,118,60,115,57,112,54,109) );
G=PermutationGroup([[(1,55,37,31,13),(2,56,38,32,14),(3,57,39,33,15),(4,58,40,34,16),(5,59,41,35,17),(6,60,42,36,18),(7,49,43,25,19),(8,50,44,26,20),(9,51,45,27,21),(10,52,46,28,22),(11,53,47,29,23),(12,54,48,30,24),(61,112,104,90,77),(62,113,105,91,78),(63,114,106,92,79),(64,115,107,93,80),(65,116,108,94,81),(66,117,97,95,82),(67,118,98,96,83),(68,119,99,85,84),(69,120,100,86,73),(70,109,101,87,74),(71,110,102,88,75),(72,111,103,89,76)], [(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)], [(1,69,10,72,7,63,4,66),(2,68,11,71,8,62,5,65),(3,67,12,70,9,61,6,64),(13,73,22,76,19,79,16,82),(14,84,23,75,20,78,17,81),(15,83,24,74,21,77,18,80),(25,92,34,95,31,86,28,89),(26,91,35,94,32,85,29,88),(27,90,36,93,33,96,30,87),(37,100,46,103,43,106,40,97),(38,99,47,102,44,105,41,108),(39,98,48,101,45,104,42,107),(49,114,58,117,55,120,52,111),(50,113,59,116,56,119,53,110),(51,112,60,115,57,118,54,109)], [(1,63,10,72,7,69,4,66),(2,68,11,65,8,62,5,71),(3,61,12,70,9,67,6,64),(13,79,22,76,19,73,16,82),(14,84,23,81,20,78,17,75),(15,77,24,74,21,83,18,80),(25,86,34,95,31,92,28,89),(26,91,35,88,32,85,29,94),(27,96,36,93,33,90,30,87),(37,106,46,103,43,100,40,97),(38,99,47,108,44,105,41,102),(39,104,48,101,45,98,42,107),(49,120,58,117,55,114,52,111),(50,113,59,110,56,119,53,116),(51,118,60,115,57,112,54,109)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30AB | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | D4 | D6 | Dic3 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×Dic3 | C5×C3⋊D4 | C4.D4 | C12.D4 | C5×C4.D4 | C5×C12.D4 |
| kernel | C5×C12.D4 | C5×C4.Dic3 | D4×C30 | C22×C30 | C12.D4 | C4.Dic3 | C6×D4 | C22×C6 | D4×C10 | C60 | C2×C20 | C22×C10 | C20 | C2×D4 | C12 | C2×C4 | C23 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C12.D4 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 15 | 227 | 0 | 0 |
| 101 | 226 | 0 | 0 |
| 225 | 15 | 0 | 225 |
| 156 | 15 | 16 | 0 |
| 240 | 0 | 224 | 0 |
| 170 | 0 | 240 | 240 |
| 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 240 | 0 | 224 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 1 | 1 | 0 |
| 71 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[15,101,225,156,227,226,15,15,0,0,0,16,0,0,225,0],[240,170,0,0,0,0,1,0,224,240,1,1,0,240,0,0],[240,0,0,71,0,0,1,0,224,240,1,1,0,1,0,0] >;
C5×C12.D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}.D_4 % in TeX
G:=Group("C5xC12.D4"); // GroupNames label
G:=SmallGroup(480,152);
// by ID
G=gap.SmallGroup(480,152);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,1410,136,4204,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations