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## G = C30.D4order 240 = 24·3·5

### 4th non-split extension by C30 of D4 acting via D4/C2=C22

Aliases: C30.4D4, C20.3D6, C151SD16, Dic62D5, D20.2S3, C12.3D10, C60.23C22, C32(Q8⋊D5), C153C87C2, C52(D4.S3), C4.16(S3×D5), (C5×Dic6)⋊4C2, (C3×D20).2C2, C6.8(C5⋊D4), C10.8(C3⋊D4), C2.5(C15⋊D4), SmallGroup(240,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C30.D4
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C30.D4
 Lower central C15 — C30 — C60 — C30.D4
 Upper central C1 — C2 — C4

Generators and relations for C30.D4
G = < a,b,c | a20=b6=1, c2=a15, bab-1=a-1, cac-1=a9, cbc-1=a5b-1 >

Character table of C30.D4

 class 1 2A 2B 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 10A 10B 12 15A 15B 20A 20B 20C 20D 20E 20F 30A 30B 60A 60B 60C 60D size 1 1 20 2 2 12 2 2 2 20 20 30 30 2 2 4 4 4 4 4 12 12 12 12 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 2 -2 0 2 2 2 0 0 0 0 2 2 -2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 -2 -1 2 0 2 2 -1 1 1 0 0 2 2 -1 -1 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ7 2 2 2 -1 2 0 2 2 -1 -1 -1 0 0 2 2 -1 -1 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 2 2 2 -1+√5/2 -1-√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 0 2 2 2 -1-√5/2 -1+√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 2 0 2 2 -2 -1+√5/2 -1-√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ11 2 2 0 2 2 -2 -1-√5/2 -1+√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ12 2 2 0 -1 -2 0 2 2 -1 √-3 -√-3 0 0 2 2 1 -1 -1 -2 -2 0 0 0 0 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ13 2 2 0 -1 -2 0 2 2 -1 -√-3 √-3 0 0 2 2 1 -1 -1 -2 -2 0 0 0 0 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ14 2 2 0 2 -2 0 -1-√5/2 -1+√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 -2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ15 2 2 0 2 -2 0 -1+√5/2 -1-√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 -2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 2 2 0 2 -2 0 -1-√5/2 -1+√5/2 2 0 0 0 0 -1-√5/2 -1+√5/2 -2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ17 2 2 0 2 -2 0 -1+√5/2 -1-√5/2 2 0 0 0 0 -1+√5/2 -1-√5/2 -2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ18 2 -2 0 2 0 0 2 2 -2 0 0 √-2 -√-2 -2 -2 0 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 complex lifted from SD16 ρ19 2 -2 0 2 0 0 2 2 -2 0 0 -√-2 √-2 -2 -2 0 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 complex lifted from SD16 ρ20 4 4 0 -2 4 0 -1-√5 -1+√5 -2 0 0 0 0 -1-√5 -1+√5 -2 1+√5/2 1-√5/2 -1-√5 -1+√5 0 0 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ21 4 4 0 -2 4 0 -1+√5 -1-√5 -2 0 0 0 0 -1+√5 -1-√5 -2 1-√5/2 1+√5/2 -1+√5 -1-√5 0 0 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ22 4 -4 0 4 0 0 -1-√5 -1+√5 -4 0 0 0 0 1+√5 1-√5 0 -1-√5 -1+√5 0 0 0 0 0 0 1-√5 1+√5 0 0 0 0 orthogonal lifted from Q8⋊D5, Schur index 2 ρ23 4 -4 0 4 0 0 -1+√5 -1-√5 -4 0 0 0 0 1-√5 1+√5 0 -1+√5 -1-√5 0 0 0 0 0 0 1+√5 1-√5 0 0 0 0 orthogonal lifted from Q8⋊D5, Schur index 2 ρ24 4 -4 0 -2 0 0 4 4 2 0 0 0 0 -4 -4 0 -2 -2 0 0 0 0 0 0 2 2 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ25 4 4 0 -2 -4 0 -1+√5 -1-√5 -2 0 0 0 0 -1+√5 -1-√5 2 1-√5/2 1+√5/2 1-√5 1+√5 0 0 0 0 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ26 4 4 0 -2 -4 0 -1-√5 -1+√5 -2 0 0 0 0 -1-√5 -1+√5 2 1+√5/2 1-√5/2 1+√5 1-√5 0 0 0 0 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ27 4 -4 0 -2 0 0 -1-√5 -1+√5 2 0 0 0 0 1+√5 1-√5 0 1+√5/2 1-√5/2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 complex faithful ρ28 4 -4 0 -2 0 0 -1-√5 -1+√5 2 0 0 0 0 1+√5 1-√5 0 1+√5/2 1-√5/2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 complex faithful ρ29 4 -4 0 -2 0 0 -1+√5 -1-√5 2 0 0 0 0 1-√5 1+√5 0 1-√5/2 1+√5/2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 complex faithful ρ30 4 -4 0 -2 0 0 -1+√5 -1-√5 2 0 0 0 0 1-√5 1+√5 0 1-√5/2 1+√5/2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 2ζ43ζ3ζ54-2ζ43ζ3ζ5+ζ43ζ54-ζ43ζ5 -2ζ4ζ3ζ53+2ζ4ζ3ζ52-ζ4ζ53+ζ4ζ52 -2ζ43ζ3ζ53+2ζ43ζ3ζ52-ζ43ζ53+ζ43ζ52 2ζ4ζ3ζ54-2ζ4ζ3ζ5+ζ4ζ54-ζ4ζ5 complex faithful

Smallest permutation representation of C30.D4
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 79 34)(2 78 35 20 80 33)(3 77 36 19 61 32)(4 76 37 18 62 31)(5 75 38 17 63 30)(6 74 39 16 64 29)(7 73 40 15 65 28)(8 72 21 14 66 27)(9 71 22 13 67 26)(10 70 23 12 68 25)(11 69 24)(41 119 100 60 120 99)(42 118 81 59 101 98)(43 117 82 58 102 97)(44 116 83 57 103 96)(45 115 84 56 104 95)(46 114 85 55 105 94)(47 113 86 54 106 93)(48 112 87 53 107 92)(49 111 88 52 108 91)(50 110 89 51 109 90)
(1 43 16 58 11 53 6 48)(2 52 17 47 12 42 7 57)(3 41 18 56 13 51 8 46)(4 50 19 45 14 60 9 55)(5 59 20 54 15 49 10 44)(21 105 36 120 31 115 26 110)(22 114 37 109 32 104 27 119)(23 103 38 118 33 113 28 108)(24 112 39 107 34 102 29 117)(25 101 40 116 35 111 30 106)(61 100 76 95 71 90 66 85)(62 89 77 84 72 99 67 94)(63 98 78 93 73 88 68 83)(64 87 79 82 74 97 69 92)(65 96 80 91 75 86 70 81)```

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

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

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

C30.D4 is a maximal subgroup of
C4014D6  C408D6  D405S3  D30.3D4  D20.34D6  C60.36D4  D20.37D6  D60.C22  D5×D4.S3  D20.9D6  D20.24D6  S3×Q8⋊D5  D20.13D6  D20.D6  D20.28D6
C30.D4 is a maximal quotient of
C30.D8  Dic6⋊Dic5  C30.SD16

Matrix representation of C30.D4 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 240 0 0 0 0 1 189 0 0 0 0 0 0 230 192 0 0 0 0 32 11
,
 15 0 0 0 0 0 20 16 0 0 0 0 0 0 189 52 0 0 0 0 240 52 0 0 0 0 0 0 1 0 0 0 0 0 93 240
,
 77 40 0 0 0 0 153 164 0 0 0 0 0 0 189 52 0 0 0 0 240 52 0 0 0 0 0 0 51 33 0 0 0 0 126 228

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,240,189,0,0,0,0,0,0,230,32,0,0,0,0,192,11],[15,20,0,0,0,0,0,16,0,0,0,0,0,0,189,240,0,0,0,0,52,52,0,0,0,0,0,0,1,93,0,0,0,0,0,240],[77,153,0,0,0,0,40,164,0,0,0,0,0,0,189,240,0,0,0,0,52,52,0,0,0,0,0,0,51,126,0,0,0,0,33,228] >;`

C30.D4 in GAP, Magma, Sage, TeX

`C_{30}.D_4`
`% in TeX`

`G:=Group("C30.D4");`
`// GroupNames label`

`G:=SmallGroup(240,16);`
`// by ID`

`G=gap.SmallGroup(240,16);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,55,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^6=1,c^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

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