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G = C60.96D4order 480 = 25·3·5

96th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.96D4, D207Dic3, Dic107Dic3, C155C4≀C2, (C2×C6).1D20, C12.7(C4×D5), (C3×D20)⋊11C4, C4○D20.2S3, (C2×C20).54D6, (C2×C30).23D4, C4.Dic34D5, C4.8(D5×Dic3), C33(D207C4), C60.124(C2×C4), (C2×C12).55D10, (C3×Dic10)⋊11C4, (C4×Dic15)⋊28C2, C53(Q83Dic3), C4.28(C15⋊D4), C12.86(C5⋊D4), C20.86(C3⋊D4), C20.24(C2×Dic3), C30.59(C22⋊C4), (C2×C60).210C22, C22.7(C3⋊D20), C6.30(D10⋊C4), C2.9(D10⋊Dic3), C10.19(C6.D4), (C3×C4○D20).5C2, (C2×C4).188(S3×D5), (C2×C10).1(C3⋊D4), (C5×C4.Dic3)⋊10C2, SmallGroup(480,52)

Series: Derived Chief Lower central Upper central

C1C60 — C60.96D4
C1C5C15C30C60C2×C60C3×C4○D20 — C60.96D4
C15C30C60 — C60.96D4
C1C4C2×C4

Generators and relations for C60.96D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a49, cbc=a45b-1 >

Subgroups: 428 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, D5, C10, C10, Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, Dic15 [×2], C60 [×2], C6×D5, C2×C30, C4×Dic5, C5×M4(2), C4○D20, Q83Dic3, C5×C3⋊C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×Dic15, C2×C60, D207C4, C5×C4.Dic3, C4×Dic15, C3×C4○D20, C60.96D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C4≀C2, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, Q83Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D207C4, D10⋊Dic3, C60.96D4

Smallest permutation representation of C60.96D4
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 46 31 16)(2 15 32 45)(3 44 33 14)(4 13 34 43)(5 42 35 12)(6 11 36 41)(7 40 37 10)(8 9 38 39)(17 30 47 60)(18 59 48 29)(19 28 49 58)(20 57 50 27)(21 26 51 56)(22 55 52 25)(23 24 53 54)(61 105)(62 74)(63 103)(64 72)(65 101)(66 70)(67 99)(69 97)(71 95)(73 93)(75 91)(76 120)(77 89)(78 118)(79 87)(80 116)(81 85)(82 114)(84 112)(86 110)(88 108)(90 106)(92 104)(94 102)(96 100)(107 119)(109 117)(111 115)
(1 98)(2 87)(3 76)(4 65)(5 114)(6 103)(7 92)(8 81)(9 70)(10 119)(11 108)(12 97)(13 86)(14 75)(15 64)(16 113)(17 102)(18 91)(19 80)(20 69)(21 118)(22 107)(23 96)(24 85)(25 74)(26 63)(27 112)(28 101)(29 90)(30 79)(31 68)(32 117)(33 106)(34 95)(35 84)(36 73)(37 62)(38 111)(39 100)(40 89)(41 78)(42 67)(43 116)(44 105)(45 94)(46 83)(47 72)(48 61)(49 110)(50 99)(51 88)(52 77)(53 66)(54 115)(55 104)(56 93)(57 82)(58 71)(59 120)(60 109)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D8A8B10A10B10C10D12A12B12C12D12E15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order122234444444455666688101010101212121212151520202020202030···3040···4060···60
size112202112203030303022242020121222442242020442222444···412···124···4

60 irreducible representations

dim111111222222222222224444444
type++++++++--++++--+
imageC1C2C2C2C4C4S3D4D4D5Dic3Dic3D6D10C3⋊D4C3⋊D4C4≀C2C4×D5C5⋊D4D20S3×D5Q83Dic3D5×Dic3C15⋊D4C3⋊D20D207C4C60.96D4
kernelC60.96D4C5×C4.Dic3C4×Dic15C3×C4○D20C3×Dic10C3×D20C4○D20C60C2×C30C4.Dic3Dic10D20C2×C20C2×C12C20C2×C10C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps111122111211122244442222248

Matrix representation of C60.96D4 in GL6(𝔽241)

52520000
1892400000
00335400
006420700
0000640
0000064
,
100000
1892400000
00345400
0023320700
0000640
000001
,
24000000
5210000
00240000
00024000
00000177
0000640

G:=sub<GL(6,GF(241))| [52,189,0,0,0,0,52,240,0,0,0,0,0,0,33,64,0,0,0,0,54,207,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[1,189,0,0,0,0,0,240,0,0,0,0,0,0,34,233,0,0,0,0,54,207,0,0,0,0,0,0,64,0,0,0,0,0,0,1],[240,52,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,64,0,0,0,0,177,0] >;

C60.96D4 in GAP, Magma, Sage, TeX

C_{60}._{96}D_4
% in TeX

G:=Group("C60.96D4");
// GroupNames label

G:=SmallGroup(480,52);
// by ID

G=gap.SmallGroup(480,52);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^49,c*b*c=a^45*b^-1>;
// generators/relations

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