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G = C60.99D4order 480 = 25·3·5

99th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.99D4, D122Dic5, C20.56D12, Dic62Dic5, C158C4≀C2, (C5×D12)⋊8C4, C20.40(C4×S3), (C5×Dic6)⋊8C4, (C4×Dic5)⋊1S3, C4○D12.1D5, (C2×C20).56D6, (C2×C30).26D4, C4.3(S3×Dic5), C55(C424S3), C60.109(C2×C4), (C12×Dic5)⋊1C2, C60.7C411C2, C10.46(D6⋊C4), (C2×C12).312D10, C32(D42Dic5), C12.64(C5⋊D4), C4.28(C5⋊D12), (C2×C60).43C22, C12.18(C2×Dic5), C6.9(C23.D5), C30.62(C22⋊C4), C2.10(D6⋊Dic5), C22.2(C15⋊D4), (C5×C4○D12).2C2, (C2×C4).137(S3×D5), (C2×C10).2(C3⋊D4), (C2×C6).47(C5⋊D4), SmallGroup(480,55)

Series: Derived Chief Lower central Upper central

C1C60 — C60.99D4
C1C5C15C30C2×C30C2×C60C12×Dic5 — C60.99D4
C15C30C60 — C60.99D4
C1C4C2×C4

Generators and relations for C60.99D4
 G = < a,b,c | a60=b4=1, c2=a15, bab-1=a49, cac-1=a29, cbc-1=a15b-1 >

Subgroups: 332 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, C12 [×2], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C5×S3, C30, C30, C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C4.Dic3, C4×C12, C4○D12, C5×Dic3, C3×Dic5 [×2], C60 [×2], S3×C10, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, C424S3, C153C8, C6×Dic5, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D42Dic5, C12×Dic5, C60.7C4, C5×C4○D12, C60.99D4
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, C424S3, S3×Dic5, C15⋊D4, C5⋊D12, D42Dic5, D6⋊Dic5, C60.99D4

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

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B10A10B10C10D10E10F10G10H12A12B12C12D12E···12L15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222344444444556668810101010101010101212121212···1215152020202020202020202030···3060···60
size1121221121010101012222226060224412121212222210···1044222244121212124···44···4

66 irreducible representations

dim111111222222222222222444444
type+++++++++--+++-+-
imageC1C2C2C2C4C4S3D4D4D5D6Dic5Dic5D10C4×S3D12C3⋊D4C4≀C2C5⋊D4C5⋊D4C424S3S3×D5S3×Dic5C5⋊D12C15⋊D4D42Dic5C60.99D4
kernelC60.99D4C12×Dic5C60.7C4C5×C4○D12C5×Dic6C5×D12C4×Dic5C60C2×C30C4○D12C2×C20Dic6D12C2×C12C20C20C2×C10C15C12C2×C6C5C2×C4C4C4C22C3C1
# reps111122111212222224448222248

Matrix representation of C60.99D4 in GL6(𝔽241)

1600000
02260000
0064000
0006400
0000151
000019051
,
100000
02400000
00240000
0006400
0000151
00000240
,
02400000
100000
0006400
00240000
0000165172
000019276

G:=sub<GL(6,GF(241))| [16,0,0,0,0,0,0,226,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,190,0,0,0,0,51,51],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,51,240],[0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,64,0,0,0,0,0,0,0,165,192,0,0,0,0,172,76] >;

C60.99D4 in GAP, Magma, Sage, TeX

C_{60}._{99}D_4
% in TeX

G:=Group("C60.99D4");
// GroupNames label

G:=SmallGroup(480,55);
// by ID

G=gap.SmallGroup(480,55);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^15,b*a*b^-1=a^49,c*a*c^-1=a^29,c*b*c^-1=a^15*b^-1>;
// generators/relations

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