Copied to
clipboard

G = Q8×D15order 240 = 24·3·5

Direct product of Q8 and D15

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

Aliases: Q8×D15, C4.6D30, C20.20D6, Dic304C2, C12.20D10, C60.6C22, C30.34C23, D30.16C22, Dic15.9C22, C53(S3×Q8), C33(Q8×D5), C158(C2×Q8), (C3×Q8)⋊2D5, (C5×Q8)⋊4S3, (Q8×C15)⋊2C2, (C4×D15).1C2, C6.34(C22×D5), C2.8(C22×D15), C10.34(C22×S3), SmallGroup(240,181)

Series: Derived Chief Lower central Upper central

C1C30 — Q8×D15
C1C5C15C30D30C4×D15 — Q8×D15
C15C30 — Q8×D15
C1C2Q8

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 [×2], C3, C4 [×3], C4 [×3], C22, C5, S3 [×2], C6, C2×C4 [×3], Q8, Q8 [×3], D5 [×2], C10, Dic3 [×3], C12 [×3], D6, C15, C2×Q8, Dic5 [×3], C20 [×3], D10, Dic6 [×3], C4×S3 [×3], C3×Q8, D15 [×2], C30, Dic10 [×3], C4×D5 [×3], C5×Q8, S3×Q8, Dic15 [×3], C60 [×3], D30, Q8×D5, Dic30 [×3], C4×D15 [×3], Q8×C15, Q8×D15
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, D10 [×3], C22×S3, D15, C22×D5, S3×Q8, D30 [×3], Q8×D5, C22×D15, Q8×D15

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

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

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

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

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

45 irreducible representations

dim11112222222444
type+++++-+++++---
imageC1C2C2C2S3Q8D5D6D10D15D30S3×Q8Q8×D5Q8×D15
kernelQ8×D15Dic30C4×D15Q8×C15C5×Q8D15C3×Q8C20C12Q8C4C5C3C1
# reps133112236412124

Matrix representation of Q8×D15 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] >;

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

׿
×
𝔽