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G = D15⋊Q8order 240 = 24·3·5

The semidirect product of D15 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D15⋊Q8, Dic64D5, C20.16D6, Dic104S3, C12.18D10, C30.7C23, Dic5.4D6, C60.28C22, Dic3.3D10, D30.10C22, Dic15.12C22, C51(S3×Q8), C31(Q8×D5), C15⋊Q83C2, C153(C2×Q8), C4.21(S3×D5), (C5×Dic6)⋊6C2, (C4×D15).3C2, C6.7(C22×D5), (C3×Dic10)⋊6C2, D30.C2.1C2, C10.7(C22×S3), (C5×Dic3).3C22, (C3×Dic5).4C22, C2.11(C2×S3×D5), SmallGroup(240,131)

Series: Derived Chief Lower central Upper central

C1C30 — D15⋊Q8
C1C5C15C30C3×Dic5D30.C2 — D15⋊Q8
C15C30 — D15⋊Q8
C1C2C4

Generators and relations for D15⋊Q8
 G = < a,b,c,d | a15=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 312 in 76 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C2×C4 [×3], Q8 [×4], D5 [×2], C10, Dic3 [×2], Dic3, C12, C12 [×2], D6, C15, C2×Q8, Dic5 [×2], Dic5, C20, C20 [×2], D10, Dic6, Dic6 [×2], C4×S3 [×3], C3×Q8, D15 [×2], C30, Dic10, Dic10 [×2], C4×D5 [×3], C5×Q8, S3×Q8, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60, D30, Q8×D5, D30.C2 [×2], C15⋊Q8 [×2], C3×Dic10, C5×Dic6, C4×D15, D15⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, D10 [×3], C22×S3, C22×D5, S3×Q8, S3×D5, Q8×D5, C2×S3×D5, D15⋊Q8

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

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

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

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

D15⋊Q8 is a maximal subgroup of
C4014D6  Dic10.D6  D30.3D4  D30.4D4  Dic10⋊D6  D30.9D4  D15⋊Q16  C60.C23  D20.38D6  C30.C24  D2024D6  C15⋊2- 1+4  D30.C23  C30.33C24  S3×Q8×D5
D15⋊Q8 is a maximal quotient of
Dic155Q8  Dic151Q8  Dic15⋊Q8  Dic156Q8  Dic15.Q8  Dic15.2Q8  Dic157Q8  D308Q8  Dic15.4Q8  D309Q8  Dic158Q8  D3010Q8  D30.Q8  D30⋊Q8  D302Q8  D303Q8  D304Q8  D30.2Q8  C20⋊Dic6

33 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B 6 10A10B12A12B12C15A15B20A20B20C20D20E20F30A30B60A60B60C60D
order1222344444455610101212121515202020202020303060606060
size11151522661010302222242020444412121212444444

33 irreducible representations

dim111111222222244444
type+++++++-+++++-+-+
imageC1C2C2C2C2C2S3Q8D5D6D6D10D10S3×Q8S3×D5Q8×D5C2×S3×D5D15⋊Q8
kernelD15⋊Q8D30.C2C15⋊Q8C3×Dic10C5×Dic6C4×D15Dic10D15Dic6Dic5C20Dic3C12C5C4C3C2C1
# reps122111122214212224

Matrix representation of D15⋊Q8 in GL6(𝔽61)

60170000
44440000
0014600
00495900
000010
000001
,
60170000
010000
0014600
0006000
0000600
0000060
,
100000
010000
0060000
0006000
00006012
0000101
,
100000
010000
001000
00496000
0000134
00004360

G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,17,44,0,0,0,0,0,0,1,49,0,0,0,0,46,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,17,1,0,0,0,0,0,0,1,0,0,0,0,0,46,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,10,0,0,0,0,12,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,49,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,34,60] >;

D15⋊Q8 in GAP, Magma, Sage, TeX

D_{15}\rtimes Q_8
% in TeX

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

G:=SmallGroup(240,131);
// by ID

G=gap.SmallGroup(240,131);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,55,218,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

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