G = C5⋊(C12.D4) order 480 = 25·3·5
The semidirect product of C5 and C12.D4 acting via C12.D4/C22×C6=C4
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊(C12.D4), C3⋊2(C23.F5), C23.2(C3⋊F5), (C22×C6).2F5, C15⋊5(C4.D4), (C22×C30).5C4, C15⋊8M4(2)⋊5C2, (C2×Dic5).74D6, (C3×Dic5).39D4, C6.26(C22⋊F5), C30.26(C22⋊C4), Dic5.8(C3⋊D4), (C22×D5).2Dic3, (C22×C10).7Dic3, C10.11(C6.D4), (C6×Dic5).169C22, C2.11(D10.D6), (D5×C2×C6).3C4, C22.5(C2×C3⋊F5), (C2×C5⋊D4).8S3, (C2×C6).42(C2×F5), (C2×C30).36(C2×C4), (C6×C5⋊D4).15C2, (C2×C10).12(C2×Dic3), SmallGroup(480,318)
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, bab-1=a-1, cac-1=dad-1=a2, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >
Subgroups: 460 in 92 conjugacy classes, 29 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, Dic5, D10, C2×C10, C2×C10, C3⋊C8, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4.D4, C5⋊C8, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C4.Dic3, C6×D4, C3×Dic5, C6×D5, C2×C30, C2×C30, C22.F5, C2×C5⋊D4, C12.D4, C15⋊C8, C6×Dic5, C3×C5⋊D4, D5×C2×C6, C22×C30, C23.F5, C15⋊8M4(2), C6×C5⋊D4, C5⋊(C12.D4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C4.D4, C2×F5, C6.D4, C3⋊F5, C22⋊F5, C12.D4, C2×C3⋊F5, C23.F5, D10.D6, C5⋊(C12.D4)
►On 120 points
Generators in S120
(1 44 24 113 36)(2 25 114 13 45)(3 46 14 115 26)(4 27 116 15 47)(5 48 16 117 28)(6 29 118 17 37)(7 38 18 119 30)(8 31 120 19 39)(9 40 20 109 32)(10 33 110 21 41)(11 42 22 111 34)(12 35 112 23 43)(49 81 93 61 103)(50 104 62 94 82)(51 83 95 63 105)(52 106 64 96 84)(53 73 85 65 107)(54 108 66 86 74)(55 75 87 67 97)(56 98 68 88 76)(57 77 89 69 99)(58 100 70 90 78)(59 79 91 71 101)(60 102 72 92 80)
(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)
(1 103 10 106 7 97 4 100)(2 102 11 105 8 108 5 99)(3 101 12 104 9 107 6 98)(13 60 22 51 19 54 16 57)(14 59 23 50 20 53 17 56)(15 58 24 49 21 52 18 55)(25 80 34 83 31 74 28 77)(26 79 35 82 32 73 29 76)(27 78 36 81 33 84 30 75)(37 88 46 91 43 94 40 85)(38 87 47 90 44 93 41 96)(39 86 48 89 45 92 42 95)(61 110 64 119 67 116 70 113)(62 109 65 118 68 115 71 112)(63 120 66 117 69 114 72 111)
(1 97 10 106 7 103 4 100)(2 102 11 99 8 108 5 105)(3 107 12 104 9 101 6 98)(13 60 22 57 19 54 16 51)(14 53 23 50 20 59 17 56)(15 58 24 55 21 52 18 49)(25 80 34 77 31 74 28 83)(26 73 35 82 32 79 29 76)(27 78 36 75 33 84 30 81)(37 88 46 85 43 94 40 91)(38 93 47 90 44 87 41 96)(39 86 48 95 45 92 42 89)(61 116 70 113 67 110 64 119)(62 109 71 118 68 115 65 112)(63 114 72 111 69 120 66 117)
G:=sub<Sym(120)| (1,44,24,113,36)(2,25,114,13,45)(3,46,14,115,26)(4,27,116,15,47)(5,48,16,117,28)(6,29,118,17,37)(7,38,18,119,30)(8,31,120,19,39)(9,40,20,109,32)(10,33,110,21,41)(11,42,22,111,34)(12,35,112,23,43)(49,81,93,61,103)(50,104,62,94,82)(51,83,95,63,105)(52,106,64,96,84)(53,73,85,65,107)(54,108,66,86,74)(55,75,87,67,97)(56,98,68,88,76)(57,77,89,69,99)(58,100,70,90,78)(59,79,91,71,101)(60,102,72,92,80), (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), (1,103,10,106,7,97,4,100)(2,102,11,105,8,108,5,99)(3,101,12,104,9,107,6,98)(13,60,22,51,19,54,16,57)(14,59,23,50,20,53,17,56)(15,58,24,49,21,52,18,55)(25,80,34,83,31,74,28,77)(26,79,35,82,32,73,29,76)(27,78,36,81,33,84,30,75)(37,88,46,91,43,94,40,85)(38,87,47,90,44,93,41,96)(39,86,48,89,45,92,42,95)(61,110,64,119,67,116,70,113)(62,109,65,118,68,115,71,112)(63,120,66,117,69,114,72,111), (1,97,10,106,7,103,4,100)(2,102,11,99,8,108,5,105)(3,107,12,104,9,101,6,98)(13,60,22,57,19,54,16,51)(14,53,23,50,20,59,17,56)(15,58,24,55,21,52,18,49)(25,80,34,77,31,74,28,83)(26,73,35,82,32,79,29,76)(27,78,36,75,33,84,30,81)(37,88,46,85,43,94,40,91)(38,93,47,90,44,87,41,96)(39,86,48,95,45,92,42,89)(61,116,70,113,67,110,64,119)(62,109,71,118,68,115,65,112)(63,114,72,111,69,120,66,117)>;
G:=Group( (1,44,24,113,36)(2,25,114,13,45)(3,46,14,115,26)(4,27,116,15,47)(5,48,16,117,28)(6,29,118,17,37)(7,38,18,119,30)(8,31,120,19,39)(9,40,20,109,32)(10,33,110,21,41)(11,42,22,111,34)(12,35,112,23,43)(49,81,93,61,103)(50,104,62,94,82)(51,83,95,63,105)(52,106,64,96,84)(53,73,85,65,107)(54,108,66,86,74)(55,75,87,67,97)(56,98,68,88,76)(57,77,89,69,99)(58,100,70,90,78)(59,79,91,71,101)(60,102,72,92,80), (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), (1,103,10,106,7,97,4,100)(2,102,11,105,8,108,5,99)(3,101,12,104,9,107,6,98)(13,60,22,51,19,54,16,57)(14,59,23,50,20,53,17,56)(15,58,24,49,21,52,18,55)(25,80,34,83,31,74,28,77)(26,79,35,82,32,73,29,76)(27,78,36,81,33,84,30,75)(37,88,46,91,43,94,40,85)(38,87,47,90,44,93,41,96)(39,86,48,89,45,92,42,95)(61,110,64,119,67,116,70,113)(62,109,65,118,68,115,71,112)(63,120,66,117,69,114,72,111), (1,97,10,106,7,103,4,100)(2,102,11,99,8,108,5,105)(3,107,12,104,9,101,6,98)(13,60,22,57,19,54,16,51)(14,53,23,50,20,59,17,56)(15,58,24,55,21,52,18,49)(25,80,34,77,31,74,28,83)(26,73,35,82,32,79,29,76)(27,78,36,75,33,84,30,81)(37,88,46,85,43,94,40,91)(38,93,47,90,44,87,41,96)(39,86,48,95,45,92,42,89)(61,116,70,113,67,110,64,119)(62,109,71,118,68,115,65,112)(63,114,72,111,69,120,66,117) );
G=PermutationGroup([[(1,44,24,113,36),(2,25,114,13,45),(3,46,14,115,26),(4,27,116,15,47),(5,48,16,117,28),(6,29,118,17,37),(7,38,18,119,30),(8,31,120,19,39),(9,40,20,109,32),(10,33,110,21,41),(11,42,22,111,34),(12,35,112,23,43),(49,81,93,61,103),(50,104,62,94,82),(51,83,95,63,105),(52,106,64,96,84),(53,73,85,65,107),(54,108,66,86,74),(55,75,87,67,97),(56,98,68,88,76),(57,77,89,69,99),(58,100,70,90,78),(59,79,91,71,101),(60,102,72,92,80)], [(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)], [(1,103,10,106,7,97,4,100),(2,102,11,105,8,108,5,99),(3,101,12,104,9,107,6,98),(13,60,22,51,19,54,16,57),(14,59,23,50,20,53,17,56),(15,58,24,49,21,52,18,55),(25,80,34,83,31,74,28,77),(26,79,35,82,32,73,29,76),(27,78,36,81,33,84,30,75),(37,88,46,91,43,94,40,85),(38,87,47,90,44,93,41,96),(39,86,48,89,45,92,42,95),(61,110,64,119,67,116,70,113),(62,109,65,118,68,115,71,112),(63,120,66,117,69,114,72,111)], [(1,97,10,106,7,103,4,100),(2,102,11,99,8,108,5,105),(3,107,12,104,9,101,6,98),(13,60,22,57,19,54,16,51),(14,53,23,50,20,59,17,56),(15,58,24,55,21,52,18,49),(25,80,34,77,31,74,28,83),(26,73,35,82,32,79,29,76),(27,78,36,75,33,84,30,81),(37,88,46,85,43,94,40,91),(38,93,47,90,44,87,41,96),(39,86,48,95,45,92,42,89),(61,116,70,113,67,110,64,119),(62,109,71,118,68,115,65,112),(63,114,72,111,69,120,66,117)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10G | 12A | 12B | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 4 | 20 | 2 | 10 | 10 | 4 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 60 | 60 | 60 | 60 | 4 | ··· | 4 | 20 | 20 | 4 | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | Dic3 | Dic3 | C3⋊D4 | F5 | C4.D4 | C2×F5 | C3⋊F5 | C22⋊F5 | C12.D4 | C2×C3⋊F5 | C23.F5 | D10.D6 | C5⋊(C12.D4) |
| kernel | C5⋊(C12.D4) | C15⋊8M4(2) | C6×C5⋊D4 | D5×C2×C6 | C22×C30 | C2×C5⋊D4 | C3×Dic5 | C2×Dic5 | C22×D5 | C22×C10 | Dic5 | C22×C6 | C15 | C2×C6 | C23 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C5⋊(C12.D4) ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 87 |
| 0 | 225 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 226 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,98,0,0,0,0,205,0,0,0,0,87],[0,16,0,0,225,0,0,0,0,0,0,226,0,0,15,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;
C5⋊(C12.D4) in GAP, Magma, Sage, TeX
C_5\rtimes (C_{12}.D_4) % in TeX
G:=Group("C5:(C12.D4)"); // GroupNames label
G:=SmallGroup(480,318);
// by ID
G=gap.SmallGroup(480,318);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^2,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations