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G = C60.29D4  order 480 = 25·3·5

29th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.29D4, C20.47D12, C12.47D20, (C2×C20).42D6, C4.Dic3⋊3D5, C4.Dic5⋊3S3, (C2×D60).11C2, (C2×C12).43D10, C12.7(C5⋊D4), C15⋊4(C4.D4), C20.6(C3⋊D4), C10.17(D6⋊C4), C5⋊2(C12.46D4), C4.12(C5⋊D12), C3⋊1(C20.46D4), C4.12(C3⋊D20), (C2×C60).91C22, (C22×D15).2C4, C6.2(D10⋊C4), C2.3(D30⋊4C4), C30.43(C22⋊C4), C22.3(D30.C2), (C2×C4).3(S3×D5), (C2×C6).1(C4×D5), (C2×C10).24(C4×S3), (C2×C30).82(C2×C4), (C3×C4.Dic5)⋊7C2, (C5×C4.Dic3)⋊7C2, SmallGroup(480,36)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C30 — C60.29D4
Chief series
C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C60.29D4
Lower central
C15 — C30 — C2×C30 — C60.29D4
Upper central
C1 — C2 — C2×C4

Generators and relations for C60.29D4
 G = < a,b,c | a60=c2=1, b4=a30, bab-1=a19, cac=a-1, cbc=a45b3 >

Subgroups: 764 in 92 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, M4(2), C2×D4, C20, D10, C2×C10, C3⋊C8, C24, D12, C2×C12, C22×S3, D15, C30, C30, C4.D4, C5⋊2C8, C40, D20, C2×C20, C22×D5, C4.Dic3, C3×M4(2), C2×D12, C60, D30, C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.46D4, C5×C3⋊C8, C3×C5⋊2C8, D60, C2×C60, C22×D15, C20.46D4, C3×C4.Dic5, C5×C4.Dic3, C2×D60, C60.29D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4.D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C12.46D4, D30.C2, C3⋊D20, C5⋊D12, C20.46D4, D30⋊4C4, C60.29D4

Smallest permutation representation of C60.29D4
►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 61 46 76 31 91 16 106)(2 80 47 95 32 110 17 65)(3 99 48 114 33 69 18 84)(4 118 49 73 34 88 19 103)(5 77 50 92 35 107 20 62)(6 96 51 111 36 66 21 81)(7 115 52 70 37 85 22 100)(8 74 53 89 38 104 23 119)(9 93 54 108 39 63 24 78)(10 112 55 67 40 82 25 97)(11 71 56 86 41 101 26 116)(12 90 57 105 42 120 27 75)(13 109 58 64 43 79 28 94)(14 68 59 83 44 98 29 113)(15 87 60 102 45 117 30 72)

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

57 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B6A6B8A8B8C8D10A10B10C10D12A12B12C15A15B20A20B20C20D20E20F24A24B24C24D30A···30F40A···40H60A···60H
order12222344556688881010101012121215152020202020202424242430···3040···4060···60
size1126060222222412122020224422444222244202020204···412···124···4

57 irreducible representations

dim111112222222222244444444
type+++++++++++++++++++
imageC1C2C2C2C4S3D4D5D6D10D12C3⋊D4C4×S3D20C5⋊D4C4×D5C4.D4S3×D5C12.46D4C3⋊D20C5⋊D12D30.C2C20.46D4C60.29D4
kernelC60.29D4C3×C4.Dic5C5×C4.Dic3C2×D60C22×D15C4.Dic5C60C4.Dic3C2×C20C2×C12C20C20C2×C10C12C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps111141221222244412222248

Matrix representation of C60.29D4 ►in GL4(𝔽241) generated by

5722000
2119100
0021184
00576
,
00443
0078197
1000
5124000
,
1000
5124000
002400
001901
G:=sub<GL(4,GF(241))| [57,21,0,0,220,191,0,0,0,0,21,57,0,0,184,6],[0,0,1,51,0,0,0,240,44,78,0,0,3,197,0,0],[1,51,0,0,0,240,0,0,0,0,240,190,0,0,0,1] >;
 

C60.29D4 in GAP, Magma, Sage, TeX 

C_{60}._{29}D_4
 
% in TeX
 
G:=Group("C60.29D4");
 
// GroupNames label
 
G:=SmallGroup(480,36);
 
// by ID
 
G=gap.SmallGroup(480,36);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,219,100,675,346,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=a^45*b^3>;
 
// generators/relations
 

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