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G = Q8xD15order 240 = 24·3·5

Direct product of Q8 and D15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8xD15, C4.6D30, C20.20D6, Dic30:4C2, C12.20D10, C60.6C22, C30.34C23, D30.16C22, Dic15.9C22, C5:3(S3xQ8), C3:3(Q8xD5), C15:8(C2xQ8), (C3xQ8):2D5, (C5xQ8):4S3, (Q8xC15):2C2, (C4xD15).1C2, C6.34(C22xD5), C2.8(C22xD15), C10.34(C22xS3), SmallGroup(240,181)

Series: Derived Chief Lower central Upper central

C1C30 — Q8xD15
C1C5C15C30D30C4xD15 — Q8xD15
C15C30 — Q8xD15
C1C2Q8

Generators and relations for Q8xD15
 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, C2xC4, Q8, Q8, D5, C10, Dic3, C12, D6, C15, C2xQ8, Dic5, C20, D10, Dic6, C4xS3, C3xQ8, D15, C30, Dic10, C4xD5, C5xQ8, S3xQ8, Dic15, C60, D30, Q8xD5, Dic30, C4xD15, Q8xC15, Q8xD15
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2xQ8, D10, C22xS3, D15, C22xD5, S3xQ8, D30, Q8xD5, C22xD15, Q8xD15

Smallest permutation representation of Q8xD15
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)]])

Q8xD15 is a maximal subgroup of
D15:SD16  D15:Q16  D20.17D6  D30.44D4  SD16:D15  Q16:D15  D20.29D6  D12.29D10  S3xQ8xD5  D20:16D6  Q8.15D30  D4.10D30
Q8xD15 is a maximal quotient of
Dic15:10Q8  C4:Dic30  Dic15.3Q8  D30:5Q8  D30:6Q8  Dic15:4Q8  D30:7Q8

45 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B 6 10A10B12A12B12C15A15B15C15D20A···20F30A30B30C30D60A···60L
order1222344444455610101212121515151520···203030303060···60
size11151522223030302222244422224···422224···4

45 irreducible representations

dim11112222222444
type+++++-+++++---
imageC1C2C2C2S3Q8D5D6D10D15D30S3xQ8Q8xD5Q8xD15
kernelQ8xD15Dic30C4xD15Q8xC15C5xQ8D15C3xQ8C20C12Q8C4C5C3C1
# reps133112236412124

Matrix representation of Q8xD15 in GL6(F61)

100000
010000
001000
000100
00006059
000011
,
100000
010000
0060000
0006000
00001738
0000244
,
43170000
4300000
000100
00606000
000010
000001
,
1600000
0600000
000100
001000
0000600
0000060

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] >;

Q8xD15 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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