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## G = Dic30⋊C4order 480 = 25·3·5

### 4th semidirect product of Dic30 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic30⋊C4
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D5×C12 — C3×D5⋊C8 — Dic30⋊C4
 Lower central C15 — C30 — C60 — Dic30⋊C4
 Upper central C1 — C2 — C4

Generators and relations for Dic30⋊C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a47, cbc-1=a45b >

Subgroups: 516 in 84 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22, C5, C6, C6 [×2], C8, C2×C4 [×3], Q8 [×3], D5 [×2], C10, Dic3 [×3], C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C24, Dic6, Dic6 [×2], C2×Dic3 [×2], C2×C12, C3×D5 [×2], C30, Q8⋊C4, C5⋊C8, Dic10 [×2], C4×D5, C4×D5, C5×Q8, C2×F5, C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C3⋊F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, C2.Dic12, C3×C5⋊C8, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, C2×C3⋊F5, Q8⋊F5, C3×D5⋊C8, C60⋊C4, D5×Dic6, Dic30⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, SD16, Q16, F5, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×F5, C24⋊C2, Dic12, D6⋊C4, C22⋊F5, C2.Dic12, S3×F5, Q8⋊F5, D6⋊F5, Dic30⋊C4

Smallest permutation representation of Dic30⋊C4
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 97 31 67)(2 96 32 66)(3 95 33 65)(4 94 34 64)(5 93 35 63)(6 92 36 62)(7 91 37 61)(8 90 38 120)(9 89 39 119)(10 88 40 118)(11 87 41 117)(12 86 42 116)(13 85 43 115)(14 84 44 114)(15 83 45 113)(16 82 46 112)(17 81 47 111)(18 80 48 110)(19 79 49 109)(20 78 50 108)(21 77 51 107)(22 76 52 106)(23 75 53 105)(24 74 54 104)(25 73 55 103)(26 72 56 102)(27 71 57 101)(28 70 58 100)(29 69 59 99)(30 68 60 98)
(2 24 50 48)(3 47 39 35)(4 10 28 22)(5 33 17 9)(6 56)(7 19 55 43)(8 42 44 30)(11 51)(12 14 60 38)(13 37 49 25)(15 23 27 59)(16 46)(18 32 54 20)(21 41)(26 36)(29 45 53 57)(34 40 58 52)(61 64 73 100)(62 87)(63 110 111 74)(65 96 89 108)(66 119 78 95)(67 82)(68 105 116 69)(70 91 94 103)(71 114 83 90)(72 77)(75 86 99 98)(76 109 88 85)(79 118 115 106)(80 81 104 93)(84 113 120 101)(92 117)(97 112)(102 107)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 8A 8B 8C 8D 10 12A 12B 12C 12D 15 20A 20B 20C 24A ··· 24H 30 60A 60B order 1 2 2 2 3 4 4 4 4 4 4 5 6 6 6 8 8 8 8 10 12 12 12 12 15 20 20 20 24 ··· 24 30 60 60 size 1 1 5 5 2 2 10 12 60 60 60 4 2 10 10 10 10 10 10 4 2 2 10 10 8 8 24 24 10 ··· 10 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 8 8 8 8 type + + + + + + + + - + - + + + + - + - image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 SD16 Q16 C3⋊D4 C4×S3 D12 C24⋊C2 Dic12 F5 C2×F5 C22⋊F5 S3×F5 Q8⋊F5 D6⋊F5 Dic30⋊C4 kernel Dic30⋊C4 C3×D5⋊C8 C60⋊C4 D5×Dic6 C5×Dic6 Dic30 D5⋊C8 C3×Dic5 C6×D5 C4×D5 C3×D5 C3×D5 Dic5 C20 D10 D5 D5 Dic6 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 4 4 1 1 2 1 1 1 2

Matrix representation of Dic30⋊C4 in GL8(𝔽241)

 240 239 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 240 0 0 0 0 0 0 1 0 240 0 0 0 0 0 1 0 0 240 0 0 0 0 1 0 0 0
,
 38 38 0 0 0 0 0 0 222 203 0 0 0 0 0 0 0 0 198 142 0 0 0 0 0 0 99 43 0 0 0 0 0 0 0 0 7 0 234 124 0 0 0 0 7 234 117 0 0 0 0 0 0 117 234 7 0 0 0 0 124 234 0 7
,
 177 0 0 0 0 0 0 0 64 64 0 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 177 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(241))| [240,1,0,0,0,0,0,0,239,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0],[38,222,0,0,0,0,0,0,38,203,0,0,0,0,0,0,0,0,198,99,0,0,0,0,0,0,142,43,0,0,0,0,0,0,0,0,7,7,0,124,0,0,0,0,0,234,117,234,0,0,0,0,234,117,234,0,0,0,0,0,124,0,7,7],[177,64,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;

Dic30⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{30}\rtimes C_4
% in TeX

G:=Group("Dic30:C4");
// GroupNames label

G:=SmallGroup(480,230);
// by ID

G=gap.SmallGroup(480,230);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,346,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c|a^60=c^4=1,b^2=a^30,b*a*b^-1=a^-1,c*a*c^-1=a^47,c*b*c^-1=a^45*b>;
// generators/relations

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