Copied to
clipboard

G = C60.29D4order 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.Dic33D5, C4.Dic53S3, (C2×D60).11C2, (C2×C12).43D10, C12.7(C5⋊D4), C154(C4.D4), C20.6(C3⋊D4), C10.17(D6⋊C4), C52(C12.46D4), C4.12(C5⋊D12), C31(C20.46D4), C4.12(C3⋊D20), (C2×C60).91C22, (C22×D15).2C4, C6.2(D10⋊C4), C2.3(D304C4), 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

C1C2×C30 — C60.29D4
C1C5C15C30C60C2×C60C3×C4.Dic5 — C60.29D4
C15C30C2×C30 — C60.29D4
C1C2C2×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 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], D5 [×2], C10, C10, C12 [×2], D6 [×4], C2×C6, C15, M4(2) [×2], C2×D4, C20 [×2], D10 [×4], C2×C10, C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], D15 [×2], C30, C30, C4.D4, C52C8, C40, D20 [×2], C2×C20, C22×D5 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], D30 [×4], C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.46D4, C5×C3⋊C8, C3×C52C8, D60 [×2], C2×C60, C22×D15 [×2], C20.46D4, C3×C4.Dic5, C5×C4.Dic3, C2×D60, C60.29D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], 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, D304C4, 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)
(1 111 46 66 31 81 16 96)(2 70 47 85 32 100 17 115)(3 89 48 104 33 119 18 74)(4 108 49 63 34 78 19 93)(5 67 50 82 35 97 20 112)(6 86 51 101 36 116 21 71)(7 105 52 120 37 75 22 90)(8 64 53 79 38 94 23 109)(9 83 54 98 39 113 24 68)(10 102 55 117 40 72 25 87)(11 61 56 76 41 91 26 106)(12 80 57 95 42 110 27 65)(13 99 58 114 43 69 28 84)(14 118 59 73 44 88 29 103)(15 77 60 92 45 107 30 62)
(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 71)(62 70)(63 69)(64 68)(65 67)(72 120)(73 119)(74 118)(75 117)(76 116)(77 115)(78 114)(79 113)(80 112)(81 111)(82 110)(83 109)(84 108)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)

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

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

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

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

׿
×
𝔽