Copied to
clipboard

## G = D4.D15order 240 = 24·3·5

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

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

 Derived series C1 — C60 — D4.D15
 Chief series C1 — C5 — C15 — C30 — C60 — Dic30 — D4.D15
 Lower central C15 — C30 — C60 — D4.D15
 Upper central C1 — C2 — C4 — D4

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 >

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

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

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

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

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 2A 2B 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 10A 10B 10C 10D 10E 10F 12 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D 30E ··· 30L 60A 60B 60C 60D order 1 2 2 3 4 4 5 5 6 6 6 8 8 10 10 10 10 10 10 12 15 15 15 15 20 20 30 30 30 30 30 ··· 30 60 60 60 60 size 1 1 4 2 2 60 2 2 2 4 4 30 30 2 2 4 4 4 4 4 2 2 2 2 4 4 2 2 2 2 4 ··· 4 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - - - image C1 C2 C2 C2 S3 D4 D5 D6 SD16 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 D4.S3 D4.D5 D4.D15 kernel D4.D15 C15⋊3C8 Dic30 D4×C15 C5×D4 C30 C3×D4 C20 C15 C12 C10 D4 C6 C4 C2 C5 C3 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 4 4 8 1 2 4

Matrix representation of D4.D15 in GL6(𝔽241)

 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 36 0 0 0 0 174 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 171 0 0 0 0 0 1 0 0 0 0 0 0 1 36 0 0 0 0 0 240
,
 240 51 0 0 0 0 190 190 0 0 0 0 0 0 15 121 0 0 0 0 0 225 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 190 190 0 0 0 0 240 51 0 0 0 0 0 0 119 11 0 0 0 0 93 122 0 0 0 0 0 0 0 202 0 0 0 0 68 0

`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

׿
×
𝔽