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G = D42D15order 240 = 24·3·5

The semidirect product of D4 and D15 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42D15, C4.5D30, C20.19D6, Dic303C2, C12.19D10, C60.5C22, C22.1D30, C30.33C23, D30.6C22, Dic15.16C22, (C3×D4)⋊3D5, (C5×D4)⋊3S3, (D4×C15)⋊3C2, (C4×D15)⋊2C2, C157D42C2, (C2×C10).3D6, (C2×C6).3D10, C1514(C4○D4), C35(D42D5), C55(D42S3), (C2×Dic15)⋊3C2, (C2×C30).1C22, C6.33(C22×D5), C2.7(C22×D15), C10.33(C22×S3), SmallGroup(240,180)

Series: Derived Chief Lower central Upper central

C1C30 — D42D15
C1C5C15C30D30C4×D15 — D42D15
C15C30 — D42D15
C1C2D4

Generators and relations for D42D15
 G = < a,b,c,d | a4=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 352 in 80 conjugacy classes, 35 normal (21 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×2], C22, C5, S3, C6, C6 [×2], C2×C4 [×3], D4, D4 [×2], Q8, D5, C10, C10 [×2], Dic3 [×3], C12, D6, C2×C6 [×2], C15, C4○D4, Dic5 [×3], C20, D10, C2×C10 [×2], Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, D15, C30, C30 [×2], Dic10, C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4, D42S3, Dic15, Dic15 [×2], C60, D30, C2×C30 [×2], D42D5, Dic30, C4×D15, C2×Dic15 [×2], C157D4 [×2], D4×C15, D42D15
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, D15, C22×D5, D42S3, D30 [×3], D42D5, C22×D15, D42D15

Smallest permutation representation of D42D15
On 120 points
Generators in S120
(1 88 17 62)(2 89 18 63)(3 90 19 64)(4 76 20 65)(5 77 21 66)(6 78 22 67)(7 79 23 68)(8 80 24 69)(9 81 25 70)(10 82 26 71)(11 83 27 72)(12 84 28 73)(13 85 29 74)(14 86 30 75)(15 87 16 61)(31 110 58 103)(32 111 59 104)(33 112 60 105)(34 113 46 91)(35 114 47 92)(36 115 48 93)(37 116 49 94)(38 117 50 95)(39 118 51 96)(40 119 52 97)(41 120 53 98)(42 106 54 99)(43 107 55 100)(44 108 56 101)(45 109 57 102)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 91)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 73)(32 74)(33 75)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 87)(47 88)(48 89)(49 90)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)
(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 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 50)(32 49)(33 48)(34 47)(35 46)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 84)(77 83)(78 82)(79 81)(85 90)(86 89)(87 88)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)

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

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

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

D42D15 is a maximal subgroup of
D30.8D4  D30.9D4  D20.10D6  D30.11D4  D8⋊D15  D83D15  SD16⋊D15  D4.5D30  C15⋊2- 1+4  D5×D42S3  S3×D42D5  D2013D6  D46D30  C4○D4×D15  D4.10D30
D42D15 is a maximal quotient of
C23.15D30  C222Dic30  C23.8D30  Dic1519D4  D30.28D4  C23.11D30  C22.D60  Dic1510Q8  Dic15.3Q8  C4.Dic30  C4⋊C47D15  D306Q8  C4⋊C4⋊D15  D4×Dic15  C23.22D30  C60.17D4  C602D4  Dic1512D4

45 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A10B10C10D10E10F 12 15A15B15C15D20A20B30A30B30C30D30E···30L60A60B60C60D
order1222234444455666101010101010121515151520203030303030···3060606060
size112230221515303022244224444422224422224···44444

45 irreducible representations

dim1111112222222222444
type+++++++++++++++---
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10D15D30D30D42S3D42D5D42D15
kernelD42D15Dic30C4×D15C2×Dic15C157D4D4×C15C5×D4C3×D4C20C2×C10C15C12C2×C6D4C4C22C5C3C1
# reps1112211212224448124

Matrix representation of D42D15 in GL6(𝔽61)

6000000
0600000
001000
000100
0000110
0000050
,
6000000
0600000
001000
000100
0000050
0000110
,
2750000
56330000
00441700
00446000
000010
000001
,
6000000
1110000
000100
001000
000010
0000060

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,50,0],[27,56,0,0,0,0,5,33,0,0,0,0,0,0,44,44,0,0,0,0,17,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;

D42D15 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_{15}
% in TeX

G:=Group("D4:2D15");
// GroupNames label

G:=SmallGroup(240,180);
// by ID

G=gap.SmallGroup(240,180);
# by ID

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

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

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