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G = C15×C8⋊C22order 480 = 25·3·5

Direct product of C15 and C8⋊C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C15×C8⋊C22, D82C30, SD161C30, C60.248D4, C12034C22, M4(2)⋊1C30, C60.296C23, C8⋊(C2×C30), C407(C2×C6), (C5×D8)⋊6C6, C246(C2×C10), C4○D44C30, (C2×D4)⋊5C30, D42(C2×C30), (C3×D8)⋊6C10, Q83(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

C1C4 — C15×C8⋊C22
C1C2C4C20C60D4×C15C15×D8 — C15×C8⋊C22
C1C2C4 — C15×C8⋊C22
C1C30C2×C60 — C15×C8⋊C22

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 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, C6, C6 [×4], C8 [×2], C2×C4, C2×C4, D4, D4 [×2], D4 [×2], Q8, C23, C10, C10 [×4], C12 [×2], C12, C2×C6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, C2×C10, C2×C10 [×5], C24 [×2], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×C6, C30, C30 [×4], C8⋊C22, C40 [×2], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×C10, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C6×D4, C3×C4○D4, C60 [×2], C60, C2×C30, C2×C30 [×5], C5×M4(2), C5×D8 [×2], C5×SD16 [×2], D4×C10, C5×C4○D4, C3×C8⋊C22, C120 [×2], C2×C60, C2×C60, D4×C15, D4×C15 [×2], D4×C15 [×2], Q8×C15, C22×C30, C5×C8⋊C22, C15×M4(2), C15×D8 [×2], C15×SD16 [×2], D4×C30, C15×C4○D4, C15×C8⋊C22
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, C2×D4, C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C8⋊C22, C5×D4 [×2], C22×C10, C6×D4, C2×C30 [×7], D4×C10, C3×C8⋊C22, D4×C15 [×2], C22×C30, C5×C8⋊C22, D4×C30, C15×C8⋊C22

Smallest permutation representation of C15×C8⋊C22
On 120 points
Generators in S120
(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 53 43 20 61 85 119 103)(2 54 44 21 62 86 120 104)(3 55 45 22 63 87 106 105)(4 56 31 23 64 88 107 91)(5 57 32 24 65 89 108 92)(6 58 33 25 66 90 109 93)(7 59 34 26 67 76 110 94)(8 60 35 27 68 77 111 95)(9 46 36 28 69 78 112 96)(10 47 37 29 70 79 113 97)(11 48 38 30 71 80 114 98)(12 49 39 16 72 81 115 99)(13 50 40 17 73 82 116 100)(14 51 41 18 74 83 117 101)(15 52 42 19 75 84 118 102)
(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 46)(29 47)(30 48)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 106)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 91)(89 92)(90 93)
(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 76)(60 77)

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), (1,53,43,20,61,85,119,103)(2,54,44,21,62,86,120,104)(3,55,45,22,63,87,106,105)(4,56,31,23,64,88,107,91)(5,57,32,24,65,89,108,92)(6,58,33,25,66,90,109,93)(7,59,34,26,67,76,110,94)(8,60,35,27,68,77,111,95)(9,46,36,28,69,78,112,96)(10,47,37,29,70,79,113,97)(11,48,38,30,71,80,114,98)(12,49,39,16,72,81,115,99)(13,50,40,17,73,82,116,100)(14,51,41,18,74,83,117,101)(15,52,42,19,75,84,118,102), (16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,46)(29,47)(30,48)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,106)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,91)(89,92)(90,93), (16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,76)(60,77)>;

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), (1,53,43,20,61,85,119,103)(2,54,44,21,62,86,120,104)(3,55,45,22,63,87,106,105)(4,56,31,23,64,88,107,91)(5,57,32,24,65,89,108,92)(6,58,33,25,66,90,109,93)(7,59,34,26,67,76,110,94)(8,60,35,27,68,77,111,95)(9,46,36,28,69,78,112,96)(10,47,37,29,70,79,113,97)(11,48,38,30,71,80,114,98)(12,49,39,16,72,81,115,99)(13,50,40,17,73,82,116,100)(14,51,41,18,74,83,117,101)(15,52,42,19,75,84,118,102), (16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,46)(29,47)(30,48)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,106)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,91)(89,92)(90,93), (16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,76)(60,77) );

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)], [(1,53,43,20,61,85,119,103),(2,54,44,21,62,86,120,104),(3,55,45,22,63,87,106,105),(4,56,31,23,64,88,107,91),(5,57,32,24,65,89,108,92),(6,58,33,25,66,90,109,93),(7,59,34,26,67,76,110,94),(8,60,35,27,68,77,111,95),(9,46,36,28,69,78,112,96),(10,47,37,29,70,79,113,97),(11,48,38,30,71,80,114,98),(12,49,39,16,72,81,115,99),(13,50,40,17,73,82,116,100),(14,51,41,18,74,83,117,101),(15,52,42,19,75,84,118,102)], [(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,46),(29,47),(30,48),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,106),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,91),(89,92),(90,93)], [(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,76),(60,77)])

165 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C5A5B5C5D6A6B6C6D6E···6J8A8B10A10B10C10D10E10F10G10H10I···10T12A12B12C12D12E12F15A···15H20A···20H20I20J20K20L24A24B24C24D30A···30H30I···30P30Q···30AN40A···40H60A···60P60Q···60X120A···120P
order12222233444555566666···688101010101010101010···1012121212121215···1520···20202020202424242430···3030···3030···3040···4060···6060···60120···120
size11244411224111111224···444111122224···42222441···12···2444444441···12···24···44···42···24···44···4

165 irreducible representations

dim111111111111111111111111222222224444
type+++++++++
imageC1C2C2C2C2C2C3C5C6C6C6C6C6C10C10C10C10C10C15C30C30C30C30C30D4D4C3×D4C3×D4C5×D4C5×D4D4×C15D4×C15C8⋊C22C3×C8⋊C22C5×C8⋊C22C15×C8⋊C22
kernelC15×C8⋊C22C15×M4(2)C15×D8C15×SD16D4×C30C15×C4○D4C5×C8⋊C22C3×C8⋊C22C5×M4(2)C5×D8C5×SD16D4×C10C5×C4○D4C3×M4(2)C3×D8C3×SD16C6×D4C3×C4○D4C8⋊C22M4(2)D8SD16C2×D4C4○D4C60C2×C30C20C2×C10C12C2×C6C4C22C15C5C3C1
# reps11221124244224884488161688112244881248

Matrix representation of C15×C8⋊C22 in GL4(𝔽241) generated by

24000
02400
00240
00024
,
240010
240001
240100
239001
,
101240
024010
0010
002240
,
100240
010240
002400
000240
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

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