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G = Q82D15order 240 = 24·3·5

The semidirect product of Q8 and D15 acting via D15/C15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82D15, C4.3D30, D60.2C2, C20.11D6, C30.36D4, C1513SD16, C12.11D10, C60.3C22, C33(Q8⋊D5), (C5×Q8)⋊3S3, (C3×Q8)⋊1D5, C153C83C2, (Q8×C15)⋊1C2, C53(Q82S3), C6.18(C5⋊D4), C2.6(C157D4), C10.18(C3⋊D4), SmallGroup(240,78)

Series: Derived Chief Lower central Upper central

C1C60 — Q82D15
C1C5C15C30C60D60 — Q82D15
C15C30C60 — Q82D15
C1C2C4Q8

Generators and relations for Q82D15
 G = < a,b,c,d | a4=c15=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

60C2
2C4
30C22
20S3
12D5
15D4
15C8
2C12
10D6
2C20
6D10
4D15
15SD16
5D12
5C3⋊C8
3D20
3C52C8
2C60
2D30
5Q82S3
3Q8⋊D5

Smallest permutation representation of Q82D15
On 120 points
Generators in S120
(1 46 16 39)(2 47 17 40)(3 48 18 41)(4 49 19 42)(5 50 20 43)(6 51 21 44)(7 52 22 45)(8 53 23 31)(9 54 24 32)(10 55 25 33)(11 56 26 34)(12 57 27 35)(13 58 28 36)(14 59 29 37)(15 60 30 38)(61 98 83 113)(62 99 84 114)(63 100 85 115)(64 101 86 116)(65 102 87 117)(66 103 88 118)(67 104 89 119)(68 105 90 120)(69 91 76 106)(70 92 77 107)(71 93 78 108)(72 94 79 109)(73 95 80 110)(74 96 81 111)(75 97 82 112)
(1 80 16 73)(2 81 17 74)(3 82 18 75)(4 83 19 61)(5 84 20 62)(6 85 21 63)(7 86 22 64)(8 87 23 65)(9 88 24 66)(10 89 25 67)(11 90 26 68)(12 76 27 69)(13 77 28 70)(14 78 29 71)(15 79 30 72)(31 117 53 102)(32 118 54 103)(33 119 55 104)(34 120 56 105)(35 106 57 91)(36 107 58 92)(37 108 59 93)(38 109 60 94)(39 110 46 95)(40 111 47 96)(41 112 48 97)(42 113 49 98)(43 114 50 99)(44 115 51 100)(45 116 52 101)
(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 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(61 91)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 113)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)

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

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

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

Q82D15 is a maximal subgroup of
D5×Q82S3  D20⋊D6  S3×Q8⋊D5  D12⋊D10  C60.39C23  D20.D6  Dic10.27D6  C60.44C23  SD16×D15  Q83D30  Q16⋊D15  D1208C2  Q8.11D30  D4⋊D30  D4.8D30
Q82D15 is a maximal quotient of
C60.2Q8  D609C4  Q82Dic15

42 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B12C15A15B15C15D20A···20F30A30B30C30D60A···60L
order1223445568810101212121515151520···203030303060···60
size116022422230302244422224···422224···4

42 irreducible representations

dim111122222222222444
type++++++++++++++
imageC1C2C2C2S3D4D5D6SD16D10C3⋊D4D15C5⋊D4D30C157D4Q82S3Q8⋊D5Q82D15
kernelQ82D15C153C8D60Q8×C15C5×Q8C30C3×Q8C20C15C12C10Q8C6C4C2C5C3C1
# reps111111212224448124

Matrix representation of Q82D15 in GL6(𝔽241)

24000000
02400000
0024019200
00123100
00002400
00000240
,
761920000
491650000
0003300
0073000
000070101
0000140171
,
190510000
1902400000
001000
000100
00002401
00002400
,
24000000
5110000
001000
0011824000
00002400
00002401

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,123,0,0,0,0,192,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[76,49,0,0,0,0,192,165,0,0,0,0,0,0,0,73,0,0,0,0,33,0,0,0,0,0,0,0,70,140,0,0,0,0,101,171],[190,190,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,51,0,0,0,0,0,1,0,0,0,0,0,0,1,118,0,0,0,0,0,240,0,0,0,0,0,0,240,240,0,0,0,0,0,1] >;

Q82D15 in GAP, Magma, Sage, TeX

Q_8\rtimes_2D_{15}
% in TeX

G:=Group("Q8:2D15");
// GroupNames label

G:=SmallGroup(240,78);
// by ID

G=gap.SmallGroup(240,78);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,55,218,116,50,964,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^15=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of Q82D15 in TeX

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