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G = D4.D15order 240 = 24·3·5

The non-split extension by D4 of D15 acting via D15/C15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D15, C4.2D30, C20.10D6, C30.35D4, C1510SD16, Dic302C2, C12.10D10, C60.2C22, C153C82C2, C33(D4.D5), C53(D4.S3), (C3×D4).1D5, (C5×D4).1S3, (D4×C15).1C2, C6.17(C5⋊D4), C2.5(C157D4), C10.17(C3⋊D4), SmallGroup(240,77)

Series: Derived Chief Lower central Upper central

C1C60 — D4.D15
C1C5C15C30C60Dic30 — D4.D15
C15C30C60 — D4.D15
C1C2C4D4

Generators and relations for D4.D15
 G = < a,b,c,d | a4=b2=c15=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
2C22
30C4
4C6
4C10
15Q8
15C8
2C2×C6
10Dic3
2C2×C10
6Dic5
4C30
15SD16
5Dic6
5C3⋊C8
3Dic10
3C52C8
2C2×C30
2Dic15
5D4.S3
3D4.D5

Smallest permutation representation of D4.D15
On 120 points
Generators in S120
(1 58 20 43)(2 59 21 44)(3 60 22 45)(4 46 23 31)(5 47 24 32)(6 48 25 33)(7 49 26 34)(8 50 27 35)(9 51 28 36)(10 52 29 37)(11 53 30 38)(12 54 16 39)(13 55 17 40)(14 56 18 41)(15 57 19 42)(61 101 90 108)(62 102 76 109)(63 103 77 110)(64 104 78 111)(65 105 79 112)(66 91 80 113)(67 92 81 114)(68 93 82 115)(69 94 83 116)(70 95 84 117)(71 96 85 118)(72 97 86 119)(73 98 87 120)(74 99 88 106)(75 100 89 107)
(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(61 90)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)
(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 86 20 72)(2 85 21 71)(3 84 22 70)(4 83 23 69)(5 82 24 68)(6 81 25 67)(7 80 26 66)(8 79 27 65)(9 78 28 64)(10 77 29 63)(11 76 30 62)(12 90 16 61)(13 89 17 75)(14 88 18 74)(15 87 19 73)(31 116 46 94)(32 115 47 93)(33 114 48 92)(34 113 49 91)(35 112 50 105)(36 111 51 104)(37 110 52 103)(38 109 53 102)(39 108 54 101)(40 107 55 100)(41 106 56 99)(42 120 57 98)(43 119 58 97)(44 118 59 96)(45 117 60 95)

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

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

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

D4.D15 is a maximal subgroup of
D5×D4.S3  C60.8C23  S3×D4.D5  C60.10C23  D1210D10  D12.24D10  D20.24D6  D2010D6  D8⋊D15  D83D15  SD16×D15  SD16⋊D15  D4.D30  D4.8D30  D4.9D30
D4.D15 is a maximal quotient of
C60.2Q8  Dic309C4  D4⋊Dic15

42 conjugacy classes

class 1 2A2B 3 4A4B5A5B6A6B6C8A8B10A10B10C10D10E10F 12 15A15B15C15D20A20B30A30B30C30D30E···30L60A60B60C60D
order1223445566688101010101010121515151520203030303030···3060606060
size1142260222443030224444422224422224···44444

42 irreducible representations

dim111122222222222444
type+++++++++++---
imageC1C2C2C2S3D4D5D6SD16D10C3⋊D4D15C5⋊D4D30C157D4D4.S3D4.D5D4.D15
kernelD4.D15C153C8Dic30D4×C15C5×D4C30C3×D4C20C15C12C10D4C6C4C2C5C3C1
# reps111111212224448124

Matrix representation of D4.D15 in GL6(𝔽241)

100000
010000
00240000
00024000
0000136
0000174240
,
100000
010000
0024017100
000100
0000136
00000240
,
240510000
1901900000
001512100
00022500
000010
000001
,
1901900000
240510000
001191100
009312200
00000202
0000680

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,36,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,171,1,0,0,0,0,0,0,1,0,0,0,0,0,36,240],[240,190,0,0,0,0,51,190,0,0,0,0,0,0,15,0,0,0,0,0,121,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[190,240,0,0,0,0,190,51,0,0,0,0,0,0,119,93,0,0,0,0,11,122,0,0,0,0,0,0,0,68,0,0,0,0,202,0] >;

D4.D15 in GAP, Magma, Sage, TeX

D_4.D_{15}
% in TeX

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

G:=SmallGroup(240,77);
// by ID

G=gap.SmallGroup(240,77);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,218,116,50,964,6917]);
// Polycyclic

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

Export

Subgroup lattice of D4.D15 in TeX

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