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G = Dic30⋊C4order 480 = 25·3·5

4th semidirect product of Dic30 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic61F5, Dic304C4, D10.19D12, D5.2Dic12, C4.2(S3×F5), C20.4(C4×S3), C60.9(C2×C4), D5⋊C8.1S3, C5⋊(C2.Dic12), (C5×Dic6)⋊4C4, (C6×D5).20D4, C31(Q8⋊F5), (C4×D5).22D6, C2.7(D6⋊F5), C12.23(C2×F5), (C3×D5).2Q16, C60⋊C4.1C2, C10.4(D6⋊C4), C152(Q8⋊C4), (D5×Dic6).5C2, (C3×D5).4SD16, C6.4(C22⋊F5), D5.2(C24⋊C2), C30.4(C22⋊C4), (C3×Dic5).23D4, Dic5.2(C3⋊D4), (D5×C12).39C22, (C3×D5⋊C8).1C2, SmallGroup(480,230)

Series: Derived Chief Lower central Upper central

C1C60 — Dic30⋊C4
C1C5C15C30C3×Dic5D5×C12C3×D5⋊C8 — Dic30⋊C4
C15C30C60 — Dic30⋊C4
C1C2C4

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 2A2B2C 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D 10 12A12B12C12D 15 20A20B20C24A···24H 30 60A60B
order122234444445666888810121212121520202024···24306060
size115522101260606042101010101010422101088242410···10888

39 irreducible representations

dim111111222222222224448888
type++++++++-+-++++-+-
imageC1C2C2C2C4C4S3D4D4D6SD16Q16C3⋊D4C4×S3D12C24⋊C2Dic12F5C2×F5C22⋊F5S3×F5Q8⋊F5D6⋊F5Dic30⋊C4
kernelDic30⋊C4C3×D5⋊C8C60⋊C4D5×Dic6C5×Dic6Dic30D5⋊C8C3×Dic5C6×D5C4×D5C3×D5C3×D5Dic5C20D10D5D5Dic6C12C6C4C3C2C1
# reps111122111122222441121112

Matrix representation of Dic30⋊C4 in GL8(𝔽241)

240239000000
11000000
0012400000
00100000
0000124000
0000102400
0000100240
00001000
,
3838000000
222203000000
001981420000
0099430000
000070234124
000072341170
000001172347
000012423407
,
1770000000
6464000000
0001770000
0017700000
00000100
00000001
00001000
00000010

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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