direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C8⋊C22, D8⋊2C30, SD16⋊1C30, C60.248D4, C120⋊34C22, M4(2)⋊1C30, C60.296C23, C8⋊(C2×C30), C40⋊7(C2×C6), (C5×D8)⋊6C6, C24⋊6(C2×C10), C4○D4⋊4C30, (C2×D4)⋊5C30, D4⋊2(C2×C30), (C3×D8)⋊6C10, Q8⋊3(C2×C30), (D4×C10)⋊14C6, (C6×D4)⋊14C10, (C15×D8)⋊14C2, (D4×C30)⋊32C2, (C5×SD16)⋊5C6, C4.14(D4×C15), C2.15(D4×C30), C12.63(C5×D4), C20.63(C3×D4), C10.78(C6×D4), C6.78(D4×C10), (C3×SD16)⋊5C10, C30.461(C2×D4), (C2×C30).131D4, (C5×M4(2))⋊5C6, C4.5(C22×C30), C22.5(D4×C15), (C15×SD16)⋊13C2, (D4×C15)⋊42C22, (C3×M4(2))⋊3C10, C20.48(C22×C6), (Q8×C15)⋊37C22, (C15×M4(2))⋊11C2, (C2×C60).440C22, C12.48(C22×C10), (C5×C4○D4)⋊11C6, (C3×C4○D4)⋊7C10, (C5×D4)⋊11(C2×C6), (C2×C4).7(C2×C30), (C5×Q8)⋊12(C2×C6), (C2×C6).24(C5×D4), (C15×C4○D4)⋊17C2, (C3×D4)⋊11(C2×C10), (C2×C20).69(C2×C6), (C3×Q8)⋊10(C2×C10), (C2×C10).25(C3×D4), (C2×C12).68(C2×C10), SmallGroup(480,941)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C8⋊C22
G = < a,b,c,d | a15=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
Subgroups: 232 in 136 conjugacy classes, 80 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, D4, Q8, C23, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C22×C6, C30, C30, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×C10, C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, C60, C60, C2×C30, C2×C30, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C3×C8⋊C22, C120, C2×C60, C2×C60, D4×C15, D4×C15, D4×C15, Q8×C15, C22×C30, C5×C8⋊C22, C15×M4(2), C15×D8, C15×SD16, D4×C30, C15×C4○D4, C15×C8⋊C22
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C2×C10, C3×D4, C22×C6, C30, C8⋊C22, C5×D4, C22×C10, C6×D4, C2×C30, D4×C10, C3×C8⋊C22, D4×C15, C22×C30, C5×C8⋊C22, D4×C30, C15×C8⋊C22
►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 47 36 93 109 80 71 22)(2 48 37 94 110 81 72 23)(3 49 38 95 111 82 73 24)(4 50 39 96 112 83 74 25)(5 51 40 97 113 84 75 26)(6 52 41 98 114 85 61 27)(7 53 42 99 115 86 62 28)(8 54 43 100 116 87 63 29)(9 55 44 101 117 88 64 30)(10 56 45 102 118 89 65 16)(11 57 31 103 119 90 66 17)(12 58 32 104 120 76 67 18)(13 59 33 105 106 77 68 19)(14 60 34 91 107 78 69 20)(15 46 35 92 108 79 70 21)
(16 89)(17 90)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 61)(42 62)(43 63)(44 64)(45 65)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 91)
(16 102)(17 103)(18 104)(19 105)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 76)(59 77)(60 78)
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), (1,47,36,93,109,80,71,22)(2,48,37,94,110,81,72,23)(3,49,38,95,111,82,73,24)(4,50,39,96,112,83,74,25)(5,51,40,97,113,84,75,26)(6,52,41,98,114,85,61,27)(7,53,42,99,115,86,62,28)(8,54,43,100,116,87,63,29)(9,55,44,101,117,88,64,30)(10,56,45,102,118,89,65,16)(11,57,31,103,119,90,66,17)(12,58,32,104,120,76,67,18)(13,59,33,105,106,77,68,19)(14,60,34,91,107,78,69,20)(15,46,35,92,108,79,70,21), (16,89)(17,90)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,61)(42,62)(43,63)(44,64)(45,65)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,91), (16,102)(17,103)(18,104)(19,105)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,76)(59,77)(60,78)>;
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), (1,47,36,93,109,80,71,22)(2,48,37,94,110,81,72,23)(3,49,38,95,111,82,73,24)(4,50,39,96,112,83,74,25)(5,51,40,97,113,84,75,26)(6,52,41,98,114,85,61,27)(7,53,42,99,115,86,62,28)(8,54,43,100,116,87,63,29)(9,55,44,101,117,88,64,30)(10,56,45,102,118,89,65,16)(11,57,31,103,119,90,66,17)(12,58,32,104,120,76,67,18)(13,59,33,105,106,77,68,19)(14,60,34,91,107,78,69,20)(15,46,35,92,108,79,70,21), (16,89)(17,90)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,61)(42,62)(43,63)(44,64)(45,65)(46,92)(47,93)(48,94)(49,95)(50,96)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,91), (16,102)(17,103)(18,104)(19,105)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,76)(59,77)(60,78) );
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)], [(1,47,36,93,109,80,71,22),(2,48,37,94,110,81,72,23),(3,49,38,95,111,82,73,24),(4,50,39,96,112,83,74,25),(5,51,40,97,113,84,75,26),(6,52,41,98,114,85,61,27),(7,53,42,99,115,86,62,28),(8,54,43,100,116,87,63,29),(9,55,44,101,117,88,64,30),(10,56,45,102,118,89,65,16),(11,57,31,103,119,90,66,17),(12,58,32,104,120,76,67,18),(13,59,33,105,106,77,68,19),(14,60,34,91,107,78,69,20),(15,46,35,92,108,79,70,21)], [(16,89),(17,90),(18,76),(19,77),(20,78),(21,79),(22,80),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,61),(42,62),(43,63),(44,64),(45,65),(46,92),(47,93),(48,94),(49,95),(50,96),(51,97),(52,98),(53,99),(54,100),(55,101),(56,102),(57,103),(58,104),(59,105),(60,91)], [(16,102),(17,103),(18,104),(19,105),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,89),(57,90),(58,76),(59,77),(60,78)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10T | 12A | 12B | 12C | 12D | 12E | 12F | 15A | ··· | 15H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 30I | ··· | 30P | 30Q | ··· | 30AN | 40A | ··· | 40H | 60A | ··· | 60P | 60Q | ··· | 60X | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | C30 | D4 | D4 | C3×D4 | C3×D4 | C5×D4 | C5×D4 | D4×C15 | D4×C15 | C8⋊C22 | C3×C8⋊C22 | C5×C8⋊C22 | C15×C8⋊C22 |
| kernel | C15×C8⋊C22 | C15×M4(2) | C15×D8 | C15×SD16 | D4×C30 | C15×C4○D4 | C5×C8⋊C22 | C3×C8⋊C22 | C5×M4(2) | C5×D8 | C5×SD16 | D4×C10 | C5×C4○D4 | C3×M4(2) | C3×D8 | C3×SD16 | C6×D4 | C3×C4○D4 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C60 | C2×C30 | C20 | C2×C10 | C12 | C2×C6 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 8 | 8 | 16 | 16 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C15×C8⋊C22 ►in GL4(𝔽241) generated by
| 24 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 240 | 0 | 1 | 0 |
| 240 | 0 | 0 | 1 |
| 240 | 1 | 0 | 0 |
| 239 | 0 | 0 | 1 |
| 1 | 0 | 1 | 240 |
| 0 | 240 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 240 |
| 1 | 0 | 0 | 240 |
| 0 | 1 | 0 | 240 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [24,0,0,0,0,24,0,0,0,0,24,0,0,0,0,24],[240,240,240,239,0,0,1,0,1,0,0,0,0,1,0,1],[1,0,0,0,0,240,0,0,1,1,1,2,240,0,0,240],[1,0,0,0,0,1,0,0,0,0,240,0,240,240,0,240] >;
C15×C8⋊C22 in GAP, Magma, Sage, TeX
C_{15}\times C_8\rtimes C_2^2 % in TeX
G:=Group("C15xC8:C2^2"); // GroupNames label
G:=SmallGroup(480,941);
// by ID
G=gap.SmallGroup(480,941);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,5126,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations