metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2F5, C120⋊2C4, C40⋊2Dic3, D10.8D12, Dic5.8Dic6, C8⋊2(C3⋊F5), C5⋊(C8⋊Dic3), C3⋊1(C40⋊C4), (C8×D5).5S3, C15⋊2(C4.Q8), C6.1(C4⋊F5), C30.8(C4⋊C4), C60.47(C2×C4), (C4×D5).86D6, (D5×C24).7C2, (C6×D5).51D4, C5⋊2C8⋊5Dic3, C12.47(C2×F5), C60⋊C4.7C2, (C3×Dic5).9Q8, (C3×D5).6SD16, C20.8(C2×Dic3), C2.4(C60⋊C4), D5.1(C24⋊C2), C10.1(C4⋊Dic3), (D5×C12).111C22, C4.8(C2×C3⋊F5), (C3×C5⋊2C8)⋊9C4, SmallGroup(480,298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C120⋊C4
G = < a,b | a120=b4=1, bab-1=a107 >
Subgroups: 460 in 72 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×Dic3, C2×C12, C3×D5, C30, C4.Q8, C5⋊2C8, C40, C4×D5, C2×F5, C4⋊Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, C4⋊F5, C8⋊Dic3, C3×C5⋊2C8, C120, D5×C12, C2×C3⋊F5, C40⋊C4, D5×C24, C60⋊C4, C120⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, SD16, F5, Dic6, D12, C2×Dic3, C4.Q8, C2×F5, C24⋊C2, C4⋊Dic3, C3⋊F5, C4⋊F5, C8⋊Dic3, C2×C3⋊F5, C40⋊C4, C60⋊C4, C120⋊C4
►On 120 points
(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 76 61 16)(2 39 110 3)(4 85 88 97)(5 48 17 84)(6 11 66 71)(7 94 115 58)(8 57 44 45)(9 20 93 32)(10 103 22 19)(12 29 120 113)(13 112 49 100)(14 75 98 87)(15 38 27 74)(18 47 54 35)(21 56 81 116)(23 102 59 90)(24 65 108 77)(25 28 37 64)(26 111 86 51)(30 83 42 119)(31 46 91 106)(33 92 69 80)(34 55 118 67)(36 101 96 41)(40 73 52 109)(43 82 79 70)(50 63 62 99)(53 72 89 60)(68 117 104 105)(78 107 114 95)
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), (1,76,61,16)(2,39,110,3)(4,85,88,97)(5,48,17,84)(6,11,66,71)(7,94,115,58)(8,57,44,45)(9,20,93,32)(10,103,22,19)(12,29,120,113)(13,112,49,100)(14,75,98,87)(15,38,27,74)(18,47,54,35)(21,56,81,116)(23,102,59,90)(24,65,108,77)(25,28,37,64)(26,111,86,51)(30,83,42,119)(31,46,91,106)(33,92,69,80)(34,55,118,67)(36,101,96,41)(40,73,52,109)(43,82,79,70)(50,63,62,99)(53,72,89,60)(68,117,104,105)(78,107,114,95)>;
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), (1,76,61,16)(2,39,110,3)(4,85,88,97)(5,48,17,84)(6,11,66,71)(7,94,115,58)(8,57,44,45)(9,20,93,32)(10,103,22,19)(12,29,120,113)(13,112,49,100)(14,75,98,87)(15,38,27,74)(18,47,54,35)(21,56,81,116)(23,102,59,90)(24,65,108,77)(25,28,37,64)(26,111,86,51)(30,83,42,119)(31,46,91,106)(33,92,69,80)(34,55,118,67)(36,101,96,41)(40,73,52,109)(43,82,79,70)(50,63,62,99)(53,72,89,60)(68,117,104,105)(78,107,114,95) );
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)], [(1,76,61,16),(2,39,110,3),(4,85,88,97),(5,48,17,84),(6,11,66,71),(7,94,115,58),(8,57,44,45),(9,20,93,32),(10,103,22,19),(12,29,120,113),(13,112,49,100),(14,75,98,87),(15,38,27,74),(18,47,54,35),(21,56,81,116),(23,102,59,90),(24,65,108,77),(25,28,37,64),(26,111,86,51),(30,83,42,119),(31,46,91,106),(33,92,69,80),(34,55,118,67),(36,101,96,41),(40,73,52,109),(43,82,79,70),(50,63,62,99),(53,72,89,60),(68,117,104,105),(78,107,114,95)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 10 | 60 | 60 | 60 | 60 | 4 | 2 | 10 | 10 | 2 | 2 | 10 | 10 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | - | + | - | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Q8 | D4 | Dic3 | Dic3 | D6 | SD16 | Dic6 | D12 | C24⋊C2 | F5 | C2×F5 | C3⋊F5 | C4⋊F5 | C2×C3⋊F5 | C40⋊C4 | C60⋊C4 | C120⋊C4 |
| kernel | C120⋊C4 | D5×C24 | C60⋊C4 | C3×C5⋊2C8 | C120 | C8×D5 | C3×Dic5 | C6×D5 | C5⋊2C8 | C40 | C4×D5 | C3×D5 | Dic5 | D10 | D5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C120⋊C4 ►in GL6(𝔽241)
| 53 | 95 | 0 | 0 | 0 | 0 |
| 208 | 150 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 229 | 126 |
| 0 | 0 | 127 | 12 | 229 | 114 |
| 0 | 0 | 12 | 127 | 0 | 114 |
| 0 | 0 | 0 | 12 | 115 | 126 |
| 223 | 186 | 0 | 0 | 0 | 0 |
| 168 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
G:=sub<GL(6,GF(241))| [53,208,0,0,0,0,95,150,0,0,0,0,0,0,12,127,12,0,0,0,0,12,127,12,0,0,229,229,0,115,0,0,126,114,114,126],[223,168,0,0,0,0,186,18,0,0,0,0,0,0,0,0,240,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,240,0,0] >;
C120⋊C4 in GAP, Magma, Sage, TeX
C_{120}\rtimes C_4 % in TeX
G:=Group("C120:C4"); // GroupNames label
G:=SmallGroup(480,298);
// by ID
G=gap.SmallGroup(480,298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,675,80,2693,14118,4724]);
// Polycyclic
G:=Group<a,b|a^120=b^4=1,b*a*b^-1=a^107>;
// generators/relations