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G = D3017D4order 480 = 25·3·5

2nd semidirect product of D30 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D3017D4, C232D30, (C2×C4)⋊2D30, (C2×C30)⋊2D4, (C6×D4)⋊14D5, (C2×D4)⋊3D15, (C2×C20)⋊21D6, C1514C22≀C2, (D4×C30)⋊26C2, (D4×C10)⋊14S3, (C2×C12)⋊21D10, C54(C232D6), C2.25(D4×D15), C6.119(D4×D5), (C22×C6)⋊4D10, (C22×C10)⋊7D6, C34(C23⋊D10), (C2×C60)⋊37C22, C30.326(C2×D4), C10.121(S3×D4), (C23×D15)⋊2C2, D303C436C2, C223(C157D4), (C22×C30)⋊3C22, C30.38D410C2, (C2×C30).308C23, (C2×Dic15)⋊2C22, C22.59(C22×D15), (C22×D15).87C22, (C2×C157D4)⋊4C2, (C2×C6)⋊5(C5⋊D4), (C2×C10)⋊9(C3⋊D4), C6.108(C2×C5⋊D4), C2.13(C2×C157D4), C10.108(C2×C3⋊D4), (C2×C6).304(C22×D5), (C2×C10).303(C22×S3), SmallGroup(480,902)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D3017D4
C1C5C15C30C2×C30C22×D15C23×D15 — D3017D4
C15C2×C30 — D3017D4
C1C22C2×D4

Generators and relations for D3017D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=c-1 >

Subgroups: 1844 in 260 conjugacy classes, 59 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×4], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], D5 [×4], C10, C10 [×2], C10 [×3], Dic3 [×2], C12, D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, Dic5 [×2], C20, D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×8], C22×C6 [×2], D15 [×4], C30, C30 [×2], C30 [×3], C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×8], C22×C10 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, Dic15 [×2], C60, D30 [×4], D30 [×12], C2×C30, C2×C30 [×2], C2×C30 [×5], D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, C232D6, C2×Dic15 [×2], C157D4 [×4], C2×C60, D4×C15 [×2], C22×D15 [×2], C22×D15 [×6], C22×C30 [×2], C23⋊D10, D303C4 [×2], C30.38D4, C2×C157D4 [×2], D4×C30, C23×D15, D3017D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, D15, C22≀C2, C5⋊D4 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, D30 [×3], D4×D5 [×2], C2×C5⋊D4, C232D6, C157D4 [×2], C22×D15, C23⋊D10, D4×D15 [×2], C2×C157D4, D3017D4

Smallest permutation representation of D3017D4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(61 71)(62 70)(63 69)(64 68)(65 67)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
(1 89 113 53)(2 90 114 54)(3 61 115 55)(4 62 116 56)(5 63 117 57)(6 64 118 58)(7 65 119 59)(8 66 120 60)(9 67 91 31)(10 68 92 32)(11 69 93 33)(12 70 94 34)(13 71 95 35)(14 72 96 36)(15 73 97 37)(16 74 98 38)(17 75 99 39)(18 76 100 40)(19 77 101 41)(20 78 102 42)(21 79 103 43)(22 80 104 44)(23 81 105 45)(24 82 106 46)(25 83 107 47)(26 84 108 48)(27 85 109 49)(28 86 110 50)(29 87 111 51)(30 88 112 52)
(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222222222344455666666610···1010···101212151515152020202030···3030···3060···60
size1111224303030302460602222244442···24···444222244442···24···44···4

84 irreducible representations

dim11111122222222222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4D15C5⋊D4D30D30C157D4S3×D4D4×D5D4×D15
kernelD3017D4D303C4C30.38D4C2×C157D4D4×C30C23×D15D4×C10D30C2×C30C6×D4C2×C20C22×C10C2×C12C22×C6C2×C10C2×D4C2×C6C2×C4C23C22C10C6C2
# reps121211142212244484816248

Matrix representation of D3017D4 in GL6(𝔽61)

6000000
0600000
00235300
0084500
000010
000001
,
6000000
010000
0018100
00434300
000010
000001
,
010000
100000
0060000
0006000
00004841
00003913
,
100000
010000
001000
000100
000010
00001760

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,23,8,0,0,0,0,53,45,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,18,43,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,48,39,0,0,0,0,41,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,60] >;

D3017D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_{17}D_4
% in TeX

G:=Group("D30:17D4");
// GroupNames label

G:=SmallGroup(480,902);
// by ID

G=gap.SmallGroup(480,902);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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