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## G = Q8×D15order 240 = 24·3·5

### Direct product of Q8 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Q8×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C4×D15 — Q8×D15
 Lower central C15 — C30 — Q8×D15
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D15
G = < a,b,c,d | a4=c15=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 336 in 76 conjugacy classes, 37 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, Q8, Q8, D5, C10, Dic3, C12, D6, C15, C2×Q8, Dic5, C20, D10, Dic6, C4×S3, C3×Q8, D15, C30, Dic10, C4×D5, C5×Q8, S3×Q8, Dic15, C60, D30, Q8×D5, Dic30, C4×D15, Q8×C15, Q8×D15
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, D10, C22×S3, D15, C22×D5, S3×Q8, D30, Q8×D5, C22×D15, Q8×D15

Smallest permutation representation of Q8×D15
On 120 points
Generators in S120
(1 50 17 43)(2 51 18 44)(3 52 19 45)(4 53 20 31)(5 54 21 32)(6 55 22 33)(7 56 23 34)(8 57 24 35)(9 58 25 36)(10 59 26 37)(11 60 27 38)(12 46 28 39)(13 47 29 40)(14 48 30 41)(15 49 16 42)(61 95 87 110)(62 96 88 111)(63 97 89 112)(64 98 90 113)(65 99 76 114)(66 100 77 115)(67 101 78 116)(68 102 79 117)(69 103 80 118)(70 104 81 119)(71 105 82 120)(72 91 83 106)(73 92 84 107)(74 93 85 108)(75 94 86 109)
(1 84 17 73)(2 85 18 74)(3 86 19 75)(4 87 20 61)(5 88 21 62)(6 89 22 63)(7 90 23 64)(8 76 24 65)(9 77 25 66)(10 78 26 67)(11 79 27 68)(12 80 28 69)(13 81 29 70)(14 82 30 71)(15 83 16 72)(31 110 53 95)(32 111 54 96)(33 112 55 97)(34 113 56 98)(35 114 57 99)(36 115 58 100)(37 116 59 101)(38 117 60 102)(39 118 46 103)(40 119 47 104)(41 120 48 105)(42 106 49 91)(43 107 50 92)(44 108 51 93)(45 109 52 94)
(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 16)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 80)(62 79)(63 78)(64 77)(65 76)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(91 107)(92 106)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)

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

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

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

Q8×D15 is a maximal subgroup of
D15⋊SD16  D15⋊Q16  D20.17D6  D30.44D4  SD16⋊D15  Q16⋊D15  D20.29D6  D12.29D10  S3×Q8×D5  D2016D6  Q8.15D30  D4.10D30
Q8×D15 is a maximal quotient of
Dic1510Q8  C4⋊Dic30  Dic15.3Q8  D305Q8  D306Q8  Dic154Q8  D307Q8

45 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6 10A 10B 12A 12B 12C 15A 15B 15C 15D 20A ··· 20F 30A 30B 30C 30D 60A ··· 60L order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 10 10 12 12 12 15 15 15 15 20 ··· 20 30 30 30 30 60 ··· 60 size 1 1 15 15 2 2 2 2 30 30 30 2 2 2 2 2 4 4 4 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + - + + + + + - - - image C1 C2 C2 C2 S3 Q8 D5 D6 D10 D15 D30 S3×Q8 Q8×D5 Q8×D15 kernel Q8×D15 Dic30 C4×D15 Q8×C15 C5×Q8 D15 C3×Q8 C20 C12 Q8 C4 C5 C3 C1 # reps 1 3 3 1 1 2 2 3 6 4 12 1 2 4

Matrix representation of Q8×D15 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 59 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 17 38 0 0 0 0 2 44
,
 43 17 0 0 0 0 43 0 0 0 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 60 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,59,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,2,0,0,0,0,38,44],[43,43,0,0,0,0,17,0,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

Q8×D15 in GAP, Magma, Sage, TeX

Q_8\times D_{15}
% in TeX

G:=Group("Q8xD15");
// GroupNames label

G:=SmallGroup(240,181);
// by ID

G=gap.SmallGroup(240,181);
# by ID

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

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

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