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G = C60.97D4order 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, D205Dic3, C12.56D20, Dic105Dic3, C156C4≀C2, (C3×D20)⋊8C4, C12.8(C4×D5), C60.94(C2×C4), (C4×Dic3)⋊1D5, C4○D20.1S3, (C2×C30).24D4, C60.7C48C2, C4.3(D5×Dic3), C33(D204C4), (Dic3×C20)⋊1C2, (C3×Dic10)⋊8C4, (C2×C20).308D6, (C2×C12).56D10, C54(Q83Dic3), C4.28(C3⋊D20), C20.61(C3⋊D4), (C2×C60).36C22, C20.39(C2×Dic3), C30.60(C22⋊C4), C22.7(C15⋊D4), C6.31(D10⋊C4), C10.20(C6.D4), C2.10(D10⋊Dic3), (C2×C4).86(S3×D5), (C3×C4○D20).2C2, (C2×C6).1(C5⋊D4), (C2×C10).47(C3⋊D4), SmallGroup(480,53)

Series: Derived Chief Lower central Upper central

C1C60 — C60.97D4
C1C5C15C30C2×C30C2×C60C3×C4○D20 — C60.97D4
C15C30C60 — C60.97D4
C1C4C2×C4

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 [×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, C20 [×2], C20 [×2], D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C4≀C2, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4.Dic3, C4×Dic3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5, C60 [×2], C6×D5, C2×C30, C4.Dic5, C4×C20, C4○D20, Q83Dic3, C153C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C10×Dic3, C2×C60, D204C4, Dic3×C20, C60.7C4, C3×C4○D20, C60.97D4
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, D204C4, 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 96 91 66)(62 77 92 107)(63 118 93 88)(64 99 94 69)(65 80 95 110)(67 102 97 72)(68 83 98 113)(70 105 100 75)(71 86 101 116)(73 108 103 78)(74 89 104 119)(76 111 106 81)(79 114 109 84)(82 117 112 87)(85 120 115 90)
(1 110 46 95 31 80 16 65)(2 79 47 64 32 109 17 94)(3 108 48 93 33 78 18 63)(4 77 49 62 34 107 19 92)(5 106 50 91 35 76 20 61)(6 75 51 120 36 105 21 90)(7 104 52 89 37 74 22 119)(8 73 53 118 38 103 23 88)(9 102 54 87 39 72 24 117)(10 71 55 116 40 101 25 86)(11 100 56 85 41 70 26 115)(12 69 57 114 42 99 27 84)(13 98 58 83 43 68 28 113)(14 67 59 112 44 97 29 82)(15 96 60 81 45 66 30 111)

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

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

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

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⋊D4C4≀C2C4×D5D20C5⋊D4D204C4S3×D5Q83Dic3D5×Dic3C3⋊D20C15⋊D4C60.97D4
kernelC60.97D4Dic3×C20C60.7C4C3×C4○D20C3×Dic10C3×D20C4○D20C60C2×C30C4×Dic3Dic10D20C2×C20C2×C12C20C2×C10C15C12C12C2×C6C3C2×C4C5C4C4C22C1
# 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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