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G = C60.97D4  order 480 = 25·3·5

97th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.97D4, D20:5Dic3, C12.56D20, Dic10:5Dic3, C15:6C4wrC2, (C3xD20):8C4, C12.8(C4xD5), C60.94(C2xC4), (C4xDic3):1D5, C4oD20.1S3, (C2xC30).24D4, C60.7C4:8C2, C4.3(D5xDic3), C3:3(D20:4C4), (Dic3xC20):1C2, (C3xDic10):8C4, (C2xC20).308D6, (C2xC12).56D10, C5:4(Q8:3Dic3), C4.28(C3:D20), C20.61(C3:D4), (C2xC60).36C22, C20.39(C2xDic3), C30.60(C22:C4), C22.7(C15:D4), C6.31(D10:C4), C10.20(C6.D4), C2.10(D10:Dic3), (C2xC4).86(S3xD5), (C3xC4oD20).2C2, (C2xC6).1(C5:D4), (C2xC10).47(C3:D4), SmallGroup(480,53)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C60 — C60.97D4
Chief series
C1 — C5 — C15 — C30 — C2xC30 — C2xC60 — C3xC4oD20 — C60.97D4
Lower central
C15 — C30 — C60 — C60.97D4
Upper central
C1 — C4 — C2xC4

Generators and relations for C60.97D4
 G = < a,b,c | a60=b4=1, c2=a45, bab-1=a41, cac-1=a29, cbc-1=a45b-1 >

Subgroups: 364 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2xC4, C2xC4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2xC6, C2xC6, C15, C42, M4(2), C4oD4, Dic5, C20, C20, D10, C2xC10, C3:C8, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C3xD5, C30, C30, C4wrC2, C5:2C8, Dic10, C4xD5, D20, C5:D4, C2xC20, C2xC20, C4.Dic3, C4xDic3, C3xC4oD4, C5xDic3, C3xDic5, C60, C6xD5, C2xC30, C4.Dic5, C4xC20, C4oD20, Q8:3Dic3, C15:3C8, C3xDic10, D5xC12, C3xD20, C3xC5:D4, C10xDic3, C2xC60, D20:4C4, Dic3xC20, C60.7C4, C3xC4oD20, C60.97D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D5, Dic3, D6, C22:C4, D10, C2xDic3, C3:D4, C4wrC2, C4xD5, D20, C5:D4, C6.D4, S3xD5, D10:C4, Q8:3Dic3, D5xDic3, C15:D4, C3:D20, D20:4C4, D10:Dic3, C60.97D4

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D8A8B10A···10F12A12B12C12D12E15A15B20A···20H20I···20X30A···30F60A···60H
order12223444444445566668810···101212121212151520···2020···2030···3060···60
size1122021126666202224202060602···22242020442···26···64···44···4

72 irreducible representations

dim111111222222222222222444444
type++++++++--++++-+-
imageC1C2C2C2C4C4S3D4D4D5Dic3Dic3D6D10C3:D4C3:D4C4wrC2C4xD5D20C5:D4D20:4C4S3xD5Q8:3Dic3D5xDic3C3:D20C15:D4C60.97D4
kernelC60.97D4Dic3xC20C60.7C4C3xC4oD20C3xDic10C3xD20C4oD20C60C2xC30C4xDic3Dic10D20C2xC20C2xC12C20C2xC10C15C12C12C2xC6C3C2xC4C5C4C4C22C1
# reps1111221112111222444416222228

Matrix representation of C60.97D4 ►in GL4(F241) generated by

240100
240000
001350
000216
,
0100
1000
002400
00064
,
024000
240000
00064
002400
G:=sub<GL(4,GF(241))| [240,240,0,0,1,0,0,0,0,0,135,0,0,0,0,216],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,64],[0,240,0,0,240,0,0,0,0,0,0,240,0,0,64,0] >;
 

C60.97D4 in GAP, Magma, Sage, TeX 

C_{60}._{97}D_4
 
% in TeX
 
G:=Group("C60.97D4");
 
// GroupNames label
 
G:=SmallGroup(480,53);
 
// by ID
 
G=gap.SmallGroup(480,53);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,675,80,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c|a^60=b^4=1,c^2=a^45,b*a*b^-1=a^41,c*a*c^-1=a^29,c*b*c^-1=a^45*b^-1>;
 
// generators/relations
 

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