Copied to
clipboard

G = D30⋊4C4order 240 = 24·3·5

2nd semidirect product of D30 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D30⋊4C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — D30⋊4C4
 Lower central C15 — C30 — D30⋊4C4
 Upper central C1 — C22

Generators and relations for D304C4
G = < a,b,c | a30=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a25b >

Subgroups: 368 in 68 conjugacy classes, 28 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4 [×2], C23, D5 [×2], C10 [×3], Dic3, C12, D6 [×4], C2×C6, C15, C22⋊C4, Dic5, C20, D10 [×4], C2×C10, C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], C2×Dic5, C2×C20, C22×D5, D6⋊C4, C5×Dic3, C3×Dic5, D30 [×2], D30 [×2], C2×C30, D10⋊C4, C6×Dic5, C10×Dic3, C22×D15, D304C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, D30.C2, C3⋊D20, C5⋊D12, D304C4

Smallest permutation representation of D304C4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
(1 106 36 71)(2 117 37 82)(3 98 38 63)(4 109 39 74)(5 120 40 85)(6 101 41 66)(7 112 42 77)(8 93 43 88)(9 104 44 69)(10 115 45 80)(11 96 46 61)(12 107 47 72)(13 118 48 83)(14 99 49 64)(15 110 50 75)(16 91 51 86)(17 102 52 67)(18 113 53 78)(19 94 54 89)(20 105 55 70)(21 116 56 81)(22 97 57 62)(23 108 58 73)(24 119 59 84)(25 100 60 65)(26 111 31 76)(27 92 32 87)(28 103 33 68)(29 114 34 79)(30 95 35 90)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D5 D6 D10 C4×S3 D12 C3⋊D4 C4×D5 D20 C5⋊D4 S3×D5 D30.C2 C3⋊D20 C5⋊D12 kernel D30⋊4C4 C6×Dic5 C10×Dic3 C22×D15 D30 C2×Dic5 C30 C2×Dic3 C2×C10 C2×C6 C10 C10 C10 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 2 1 2 2 2 2 4 4 4 2 2 2 2

Matrix representation of D304C4 in GL4(𝔽61) generated by

 0 1 0 0 60 60 0 0 0 0 43 1 0 0 42 1
,
 0 1 0 0 1 0 0 0 0 0 0 44 0 0 43 0
,
 50 0 0 0 11 11 0 0 0 0 25 7 0 0 50 36
`G:=sub<GL(4,GF(61))| [0,60,0,0,1,60,0,0,0,0,43,42,0,0,1,1],[0,1,0,0,1,0,0,0,0,0,0,43,0,0,44,0],[50,11,0,0,0,11,0,0,0,0,25,50,0,0,7,36] >;`

D304C4 in GAP, Magma, Sage, TeX

`D_{30}\rtimes_4C_4`
`% in TeX`

`G:=Group("D30:4C4");`
`// GroupNames label`

`G:=SmallGroup(240,28);`
`// by ID`

`G=gap.SmallGroup(240,28);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,79,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;`
`// generators/relations`

׿
×
𝔽