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G = D15⋊D8order 480 = 25·3·5

The semidirect product of D15 and D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D204D6, D152D8, D124D10, D30.37D4, D603C22, C60.5C23, Dic15.11D4, C53(S3×D8), C33(D5×D8), C159(C2×D8), D4⋊S32D5, D4⋊D52S3, C3⋊C814D10, (C5×D4)⋊3D6, D45(S3×D5), (C3×D4)⋊3D10, (D4×D15)⋊1C2, C3⋊D402C2, C52C814D6, C5⋊D242C2, C6.67(D4×D5), C20⋊D61C2, C10.68(S3×D4), D152C81C2, C30.167(C2×D4), (D4×C15)⋊5C22, (C5×D12)⋊3C22, (C3×D20)⋊3C22, C20.5(C22×S3), C12.5(C22×D5), (C4×D15).1C22, C2.20(D10⋊D6), C4.5(C2×S3×D5), (C5×D4⋊S3)⋊3C2, (C3×D4⋊D5)⋊3C2, (C5×C3⋊C8)⋊3C22, (C3×C52C8)⋊3C22, SmallGroup(480,557)

Series: Derived Chief Lower central Upper central

C1C60 — D15⋊D8
C1C5C15C30C60C3×D20C20⋊D6 — D15⋊D8
C15C30C60 — D15⋊D8
C1C2C4D4

Generators and relations for D15⋊D8
 G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1228 in 152 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, D4, D4 [×5], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6 [×7], C2×C6 [×2], C15, C2×C8, D8 [×4], C2×D4 [×2], Dic5, C20, D10 [×7], C2×C10 [×2], C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3 [×2], C5×S3, C3×D5, D15 [×2], D15, C30, C30, C2×D8, C52C8, C40, C4×D5, D20, D20, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5 [×2], S3×C8, D24, D4⋊S3, D4⋊S3, C3×D8, S3×D4 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, D30 [×3], C2×C30, C8×D5, D40, D4⋊D5, D4⋊D5, C5×D8, D4×D5 [×2], S3×D8, C5×C3⋊C8, C3×C52C8, C15⋊D4, C3×D20, C5×D12, C4×D15, D60, C157D4, D4×C15, C2×S3×D5, C22×D15, D5×D8, D152C8, C3⋊D40, C5⋊D24, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D4×D15, D15⋊D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C22×S3, C2×D8, C22×D5, S3×D4, S3×D5, D4×D5, S3×D8, C2×S3×D5, D5×D8, D10⋊D6, D15⋊D8

Smallest permutation representation of D15⋊D8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(91 98)(92 97)(93 96)(94 95)(99 105)(100 104)(101 103)(106 114)(107 113)(108 112)(109 111)(115 120)(116 119)(117 118)
(1 69 32 118 20 80 54 95)(2 65 33 114 21 76 55 91)(3 61 34 110 22 87 56 102)(4 72 35 106 23 83 57 98)(5 68 36 117 24 79 58 94)(6 64 37 113 25 90 59 105)(7 75 38 109 26 86 60 101)(8 71 39 120 27 82 46 97)(9 67 40 116 28 78 47 93)(10 63 41 112 29 89 48 104)(11 74 42 108 30 85 49 100)(12 70 43 119 16 81 50 96)(13 66 44 115 17 77 51 92)(14 62 45 111 18 88 52 103)(15 73 31 107 19 84 53 99)
(2 12)(3 8)(5 15)(6 11)(9 14)(16 21)(18 28)(19 24)(22 27)(25 30)(31 58)(32 54)(33 50)(34 46)(35 57)(36 53)(37 49)(38 60)(39 56)(40 52)(41 48)(42 59)(43 55)(44 51)(45 47)(61 97)(62 93)(63 104)(64 100)(65 96)(66 92)(67 103)(68 99)(69 95)(70 91)(71 102)(72 98)(73 94)(74 105)(75 101)(76 119)(77 115)(78 111)(79 107)(80 118)(81 114)(82 110)(83 106)(84 117)(85 113)(86 109)(87 120)(88 116)(89 112)(90 108)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F 12 15A15B20A20B24A24B30A30B30C30D30E30F40A40B40C40D60A60B
order1222222234455666888810101010101012151520202424303030303030404040406060
size11412151520602230222840661010228824244444420204488881212121288

45 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D8D10D10D10S3×D4S3×D5D4×D5S3×D8C2×S3×D5D5×D8D10⋊D6D15⋊D8
kernelD15⋊D8D152C8C3⋊D40C5⋊D24C3×D4⋊D5C5×D4⋊S3C20⋊D6D4×D15D4⋊D5Dic15D30D4⋊S3C52C8D20C5×D4D15C3⋊C8D12C3×D4C10D4C6C5C4C3C2C1
# reps111111111112111422212222442

Matrix representation of D15⋊D8 in GL6(𝔽241)

2401890000
52520000
0015400
0017423900
000010
000001
,
1520000
02400000
0024018700
000100
00002400
00000240
,
24000000
02400000
001000
0017424000
0000022
000023022
,
24000000
02400000
001000
0017424000
000010
00001240

G:=sub<GL(6,GF(241))| [240,52,0,0,0,0,189,52,0,0,0,0,0,0,1,174,0,0,0,0,54,239,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,52,240,0,0,0,0,0,0,240,0,0,0,0,0,187,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,0,230,0,0,0,0,22,22],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,240] >;

D15⋊D8 in GAP, Magma, Sage, TeX

D_{15}\rtimes D_8
% in TeX

G:=Group("D15:D8");
// GroupNames label

G:=SmallGroup(480,557);
// by ID

G=gap.SmallGroup(480,557);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,675,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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