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G = D6013C4order 480 = 25·3·5

7th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6013C4, C12.57D20, C60.100D4, Dic3013C4, C159C4≀C2, C12.9(C4×D5), C20.65(C4×S3), C60.95(C2×C4), (C4×Dic3)⋊2D5, (C2×C30).27D4, C4.Dic51S3, C31(D204C4), C53(D12⋊C4), (Dic3×C20)⋊2C2, (C2×C20).309D6, (C2×C12).58D10, (C2×C10).33D12, C10.23(D6⋊C4), C4.29(C3⋊D20), C20.62(C3⋊D4), C4.3(D30.C2), (C2×C60).37C22, D6011C2.4C2, C6.8(D10⋊C4), C2.9(D304C4), C30.63(C22⋊C4), C22.2(C5⋊D12), (C2×C4).87(S3×D5), (C2×C6).3(C5⋊D4), (C3×C4.Dic5)⋊4C2, SmallGroup(480,56)

Series: Derived Chief Lower central Upper central

C1C60 — D6013C4
C1C5C15C30C60C2×C60C3×C4.Dic5 — D6013C4
C15C30C60 — D6013C4
C1C4C2×C4

Generators and relations for D6013C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a41, cbc-1=a55b >

Subgroups: 508 in 88 conjugacy classes, 32 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, D5, C10, C10, Dic3 [×3], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C4≀C2, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3 [×2], Dic15, C60 [×2], D30, C2×C30, C4.Dic5, C4×C20, C4○D20, D12⋊C4, C3×C52C8, C10×Dic3, Dic30, C4×D15, D60, C157D4, C2×C60, D204C4, C3×C4.Dic5, Dic3×C20, D6011C2, D6013C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, D12⋊C4, D30.C2, C3⋊D20, C5⋊D12, D204C4, D304C4, D6013C4

Smallest permutation representation of D6013C4
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 61)(2 120)(3 119)(4 118)(5 117)(6 116)(7 115)(8 114)(9 113)(10 112)(11 111)(12 110)(13 109)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 101)(22 100)(23 99)(24 98)(25 97)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
(1 46 31 16)(2 27 32 57)(3 8 33 38)(4 49 34 19)(5 30 35 60)(6 11 36 41)(7 52 37 22)(9 14 39 44)(10 55 40 25)(12 17 42 47)(13 58 43 28)(15 20 45 50)(18 23 48 53)(21 26 51 56)(24 29 54 59)(61 111)(62 92)(63 73)(64 114)(65 95)(66 76)(67 117)(68 98)(69 79)(70 120)(71 101)(72 82)(74 104)(75 85)(77 107)(78 88)(80 110)(81 91)(83 113)(84 94)(86 116)(87 97)(89 119)(90 100)(93 103)(96 106)(99 109)(102 112)(105 115)(108 118)

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B8A8B10A···10F12A12B12C15A15B20A···20H20I···20X24A24B24C24D30A···30F60A···60H
order122234444444455668810···10121212151520···2020···202424242430···3060···60
size112602112666660222420202···2224442···26···6202020204···44···4

72 irreducible representations

dim11111122222222222222444444
type++++++++++++++++
imageC1C2C2C2C4C4S3D4D4D5D6D10C4×S3C3⋊D4D12C4≀C2C4×D5D20C5⋊D4D204C4S3×D5D12⋊C4D30.C2C3⋊D20C5⋊D12D6013C4
kernelD6013C4C3×C4.Dic5Dic3×C20D6011C2Dic30D60C4.Dic5C60C2×C30C4×Dic3C2×C20C2×C12C20C20C2×C10C15C12C12C2×C6C3C2×C4C5C4C4C22C1
# reps111122111212222444416222228

Matrix representation of D6013C4 in GL4(𝔽241) generated by

6000
020100
00244
00104240
,
09100
98000
00018
00670
,
64000
024000
002400
001371
G:=sub<GL(4,GF(241))| [6,0,0,0,0,201,0,0,0,0,2,104,0,0,44,240],[0,98,0,0,91,0,0,0,0,0,0,67,0,0,18,0],[64,0,0,0,0,240,0,0,0,0,240,137,0,0,0,1] >;

D6013C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{13}C_4
% in TeX

G:=Group("D60:13C4");
// GroupNames label

G:=SmallGroup(480,56);
// by ID

G=gap.SmallGroup(480,56);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,100,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^55*b>;
// generators/relations

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