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G = C60.36D4order 480 = 25·3·5

36th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.36D4, D1221D10, D20.35D6, Dic618D10, C60.151C23, C4○D123D5, (C2×D20)⋊8S3, (C6×D20)⋊1C2, C15⋊D813C2, C157(C8⋊C22), (C2×C20).85D6, (C2×C30).41D4, C30.73(C2×D4), C60.7C46C2, C34(D4⋊D10), (C2×C12).86D10, C54(D126C22), C153C821C22, C30.D413C2, (C5×D12)⋊28C22, C4.23(C15⋊D4), C12.26(C5⋊D4), C20.81(C3⋊D4), (C2×C60).31C22, C20.86(C22×S3), C12.86(C22×D5), (C5×Dic6)⋊24C22, (C3×D20).41C22, C22.4(C15⋊D4), (C2×C4).7(S3×D5), C4.124(C2×S3×D5), (C5×C4○D12)⋊1C2, C6.73(C2×C5⋊D4), C2.7(C2×C15⋊D4), C10.74(C2×C3⋊D4), (C2×C6).52(C5⋊D4), (C2×C10).9(C3⋊D4), SmallGroup(480,374)

Series: Derived Chief Lower central Upper central

C1C60 — C60.36D4
C1C5C15C30C60C3×D20C15⋊D8 — C60.36D4
C15C30C60 — C60.36D4
C1C2C2×C4

Generators and relations for C60.36D4
 G = < a,b,c | a60=c2=1, b4=a30, bab-1=a-1, cac=a19, cbc=a30b3 >

Subgroups: 668 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×3], C22×C6, C5×S3, C3×D5 [×2], C30, C30, C8⋊C22, C52C8 [×2], D20 [×2], D20, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C4○D12, C6×D4, C5×Dic3, C60 [×2], C6×D5 [×4], S3×C10, C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, D126C22, C153C8 [×2], C3×D20 [×2], C3×D20, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D5×C2×C6, D4⋊D10, C15⋊D8 [×2], C30.D4 [×2], C60.7C4, C6×D20, C5×C4○D12, C60.36D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, D126C22, C15⋊D4 [×2], C2×S3×D5, D4⋊D10, C2×C15⋊D4, C60.36D4

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

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D6E6F6G8A8B10A10B10C10D10E10F10G10H12A12B15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222223444556666666881010101010101010121215152020202020202020202030···3060···60
size11212202022212222222020202060602244121212124444222244121212124···44···4

57 irreducible representations

dim111111222222222222244444444
type+++++++++++++++++-+-+
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D10C3⋊D4C3⋊D4C5⋊D4C5⋊D4C8⋊C22S3×D5D126C22C15⋊D4C2×S3×D5C15⋊D4D4⋊D10C60.36D4
kernelC60.36D4C15⋊D8C30.D4C60.7C4C6×D20C5×C4○D12C2×D20C60C2×C30C4○D12D20C2×C20Dic6D12C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps122111111221222224412222248

Matrix representation of C60.36D4 in GL6(𝔽241)

22500000
189150000
007816300
007819700
000078163
000078197
,
30470000
2272110000
0078821329
00204163234109
009383163159
0061483778
,
24000000
02400000
001000
005124000
00124188443
00511778197

G:=sub<GL(6,GF(241))| [225,189,0,0,0,0,0,15,0,0,0,0,0,0,78,78,0,0,0,0,163,197,0,0,0,0,0,0,78,78,0,0,0,0,163,197],[30,227,0,0,0,0,47,211,0,0,0,0,0,0,78,204,93,6,0,0,82,163,83,148,0,0,132,234,163,37,0,0,9,109,159,78],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,51,124,5,0,0,0,240,188,117,0,0,0,0,44,78,0,0,0,0,3,197] >;

C60.36D4 in GAP, Magma, Sage, TeX

C_{60}._{36}D_4
% in TeX

G:=Group("C60.36D4");
// GroupNames label

G:=SmallGroup(480,374);
// by ID

G=gap.SmallGroup(480,374);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^-1,c*a*c=a^19,c*b*c=a^30*b^3>;
// generators/relations

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