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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×C8⋊C22
 Chief series C1 — C2 — C4 — C20 — C60 — D4×C15 — C15×D8 — C15×C8⋊C22
 Lower central C1 — C2 — C4 — C15×C8⋊C22
 Upper central C1 — C30 — C2×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 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

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