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G = D607C4order 480 = 25·3·5

1st semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D607C4, C4.17D60, C424D15, Dic307C4, C12.35D20, C20.35D12, C60.166D4, C1515C4≀C2, (C4×C20)⋊8S3, (C4×C60)⋊8C2, (C4×C12)⋊6D5, C4.6(C4×D15), C20.81(C4×S3), C12.49(C4×D5), (C2×C4).69D30, C60.7C41C2, C32(D204C4), C54(C424S3), C60.186(C2×C4), (C2×C20).397D6, (C2×C30).133D4, C10.27(D6⋊C4), (C2×C12).402D10, D6011C2.1C2, C2.3(D303C4), C30.69(C22⋊C4), (C2×C60).483C22, C22.7(C157D4), C6.12(D10⋊C4), (C2×C6).65(C5⋊D4), (C2×C10).65(C3⋊D4), SmallGroup(480,165)

Series: Derived Chief Lower central Upper central

C1C60 — D607C4
C1C5C15C30C2×C30C2×C60D6011C2 — D607C4
C15C30C60 — D607C4
C1C4C2×C4C42

Generators and relations for D607C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >

Subgroups: 516 in 88 conjugacy classes, 33 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, C12 [×2], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, D15, C30, C30, C4≀C2, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4.Dic3, C4×C12, C4○D12, Dic15, C60 [×2], C60 [×2], D30, C2×C30, C4.Dic5, C4×C20, C4○D20, C424S3, C153C8, Dic30, C4×D15, D60, C157D4, C2×C60, C2×C60, D204C4, C60.7C4, C4×C60, D6011C2, D607C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C424S3, C4×D15, D60, C157D4, D204C4, D303C4, D607C4

Smallest permutation representation of D607C4
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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 120)(11 119)(12 118)(13 117)(14 116)(15 115)(16 114)(17 113)(18 112)(19 111)(20 110)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 106 91 76)(62 107 92 77)(63 108 93 78)(64 109 94 79)(65 110 95 80)(66 111 96 81)(67 112 97 82)(68 113 98 83)(69 114 99 84)(70 115 100 85)(71 116 101 86)(72 117 102 87)(73 118 103 88)(74 119 104 89)(75 120 105 90)

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

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

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

126 conjugacy classes

class 1 2A2B2C 3 4A4B4C···4G4H5A5B6A6B6C8A8B10A···10F12A···12L15A15B15C15D20A···20X30A···30L60A···60AV
order12223444···44556668810···1012···121515151520···2030···3060···60
size112602112···2602222260602···22···222222···22···22···2

126 irreducible representations

dim111111222222222222222222222
type+++++++++++++++
imageC1C2C2C2C4C4S3D4D4D5D6D10C4×S3D12C3⋊D4D15C4≀C2C4×D5D20C5⋊D4D30C424S3C4×D15D60C157D4D204C4D607C4
kernelD607C4C60.7C4C4×C60D6011C2Dic30D60C4×C20C60C2×C30C4×C12C2×C20C2×C12C20C20C2×C10C42C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps11112211121222244444488881632

Matrix representation of D607C4 in GL2(𝔽241) generated by

1180
096
,
096
1180
,
2400
0177
G:=sub<GL(2,GF(241))| [118,0,0,96],[0,118,96,0],[240,0,0,177] >;

D607C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_7C_4
% in TeX

G:=Group("D60:7C4");
// GroupNames label

G:=SmallGroup(480,165);
// by ID

G=gap.SmallGroup(480,165);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,675,80,2693,18822]);
// Polycyclic

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

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