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G = C60.98D4order 480 = 25·3·5

98th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.98D4, D124Dic5, Dic64Dic5, C157C4≀C2, (C5×D12)⋊11C4, C20.39(C4×S3), C4○D12.2D5, (C2×C10).1D12, (C2×C30).25D4, (C2×C20).55D6, C4.Dic54S3, C4.8(S3×Dic5), C55(D12⋊C4), C60.125(C2×C4), (C5×Dic6)⋊11C4, (C2×C12).57D10, C12.3(C2×Dic5), C10.45(D6⋊C4), (C4×Dic15)⋊29C2, C31(D42Dic5), C4.29(C15⋊D4), C12.87(C5⋊D4), C20.87(C3⋊D4), C2.9(D6⋊Dic5), C6.8(C23.D5), C30.61(C22⋊C4), (C2×C60).211C22, C22.7(C5⋊D12), (C5×C4○D12).5C2, (C2×C4).189(S3×D5), (C2×C6).2(C5⋊D4), (C3×C4.Dic5)⋊10C2, SmallGroup(480,54)

Series: Derived Chief Lower central Upper central

C1C60 — C60.98D4
C1C5C15C30C60C2×C60C3×C4.Dic5 — C60.98D4
C15C30C60 — C60.98D4
C1C4C2×C4

Generators and relations for C60.98D4
 G = < a,b,c,d | a12=c10=1, b2=a6, d2=c5, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 380 in 88 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C10, C10 [×2], Dic3 [×3], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3, Dic15 [×2], C60 [×2], S3×C10, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, D12⋊C4, C3×C52C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×Dic15, C2×C60, D42Dic5, C3×C4.Dic5, C4×Dic15, C5×C4○D12, C60.98D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C4≀C2, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, D12⋊C4, S3×Dic5, C15⋊D4, C5⋊D12, D42Dic5, D6⋊Dic5, C60.98D4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B8A8B10A10B10C10D10E10F10G10H12A12B12C15A15B20A20B20C20D20E20F20G20H20I20J24A24B24C24D30A···30F60A···60H
order122234444444455668810101010101010101212121515202020202020202020202424242430···3060···60
size1121221121230303030222420202244121212122244422224412121212202020204···44···4

60 irreducible representations

dim111111222222222222224444444
type+++++++++--+++--+
imageC1C2C2C2C4C4S3D4D4D5D6Dic5Dic5D10C4×S3C3⋊D4D12C4≀C2C5⋊D4C5⋊D4S3×D5D12⋊C4S3×Dic5C15⋊D4C5⋊D12D42Dic5C60.98D4
kernelC60.98D4C3×C4.Dic5C4×Dic15C5×C4○D12C5×Dic6C5×D12C4.Dic5C60C2×C30C4○D12C2×C20Dic6D12C2×C12C20C20C2×C10C15C12C2×C6C2×C4C5C4C4C22C3C1
# reps111122111212222224442222248

Matrix representation of C60.98D4 in GL6(𝔽241)

24000000
02400000
0064000
00017700
00002401
00002400
,
76890000
1031650000
00017700
00177000
000001
000010
,
2401910000
2401900000
001000
00024000
000010
000001
,
511910000
521900000
00240000
00017700
000001
000010

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[76,103,0,0,0,0,89,165,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[240,240,0,0,0,0,191,190,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,52,0,0,0,0,191,190,0,0,0,0,0,0,240,0,0,0,0,0,0,177,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C60.98D4 in GAP, Magma, Sage, TeX

C_{60}._{98}D_4
% in TeX

G:=Group("C60.98D4");
// GroupNames label

G:=SmallGroup(480,54);
// by ID

G=gap.SmallGroup(480,54);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^10=1,b^2=a^6,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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