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G = C60.38D4order 480 = 25·3·5

38th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.38D4, D1219D10, C20.51D12, Dic616D10, C60.129C23, D60.49C22, C52C84D6, C4○D124D5, C55(C8⋊D6), (C2×D60)⋊18C2, C5⋊D2414C2, (C2×C20).91D6, (C2×C10).6D12, C30.80(C2×D4), (C2×C30).48D4, C4.Dic58S3, C31(D4⋊D10), C1511(C8⋊C22), C10.50(C2×D12), (C2×C12).91D10, Dic6⋊D513C2, (C5×D12)⋊21C22, C12.28(C5⋊D4), C4.16(C5⋊D12), C20.91(C22×S3), (C2×C60).94C22, (C5×Dic6)⋊18C22, C12.152(C22×D5), C22.9(C5⋊D12), C4.77(C2×S3×D5), C6.4(C2×C5⋊D4), (C5×C4○D12)⋊6C2, (C2×C4).11(S3×D5), C2.8(C2×C5⋊D12), (C3×C4.Dic5)⋊8C2, (C3×C52C8)⋊18C22, (C2×C6).12(C5⋊D4), SmallGroup(480,381)

Series: Derived Chief Lower central Upper central

C1C60 — C60.38D4
C1C5C15C30C60C3×C52C8C5⋊D24 — C60.38D4
C15C30C60 — C60.38D4
C1C2C2×C4

Generators and relations for C60.38D4
 G = < a,b,c | a60=c2=1, b4=a30, bab-1=a19, cac=a-1, cbc=a30b3 >

Subgroups: 956 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12 [×2], D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C24 [×2], Dic6, C4×S3, D12, D12 [×3], C3⋊D4, C2×C12, C22×S3, C5×S3, D15 [×2], C30, C30, C8⋊C22, C52C8 [×2], D20 [×3], C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C22×D5, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, C4○D12, C5×Dic3, C60 [×2], S3×C10, D30 [×4], C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, C8⋊D6, C3×C52C8 [×2], C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, D60 [×2], D60, C2×C60, C22×D15, D4⋊D10, C5⋊D24 [×2], Dic6⋊D5 [×2], C3×C4.Dic5, C5×C4○D12, C2×D60, C60.38D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8⋊D6, C5⋊D12 [×2], C2×S3×D5, D4⋊D10, C2×C5⋊D12, C60.38D4

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

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D10E10F10G10H12A12B12C15A15B20A20B20C20D20E20F20G20H20I20J24A24B24C24D30A···30F60A···60H
order122222344455668810101010101010101212121515202020202020202020202424242430···3060···60
size11212606022212222420202244121212122244422224412121212202020204···44···4

57 irreducible representations

dim111111222222222222244444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D10D12D12C5⋊D4C5⋊D4C8⋊C22S3×D5C8⋊D6C5⋊D12C2×S3×D5C5⋊D12D4⋊D10C60.38D4
kernelC60.38D4C5⋊D24Dic6⋊D5C3×C4.Dic5C5×C4○D12C2×D60C4.Dic5C60C2×C30C4○D12C52C8C2×C20Dic6D12C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps122111111221222224412222248

Matrix representation of C60.38D4 in GL4(𝔽241) generated by

1842100
2205000
222222057
219180184235
,
214157153235
226278588
162122784
6422515214
,
1000
5124000
12042443
13712178197
G:=sub<GL(4,GF(241))| [184,220,22,219,21,50,22,180,0,0,220,184,0,0,57,235],[214,226,16,64,157,27,212,225,153,85,27,15,235,88,84,214],[1,51,120,137,0,240,42,121,0,0,44,78,0,0,3,197] >;

C60.38D4 in GAP, Magma, Sage, TeX

C_{60}._{38}D_4
% in TeX

G:=Group("C60.38D4");
// GroupNames label

G:=SmallGroup(480,381);
// by ID

G=gap.SmallGroup(480,381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,100,346,80,1356,18822]);
// Polycyclic

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

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