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G = D304D4order 480 = 25·3·5

4th semidirect product of D30 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D304D4, D104D12, (C6×D5)⋊4D4, C152C22≀C2, (C2×D12)⋊6D5, (S3×C10)⋊4D4, (C2×C12)⋊1D10, (C2×C20)⋊18D6, C6.26(D4×D5), D65(C5⋊D4), C53(D6⋊D4), (C2×Dic5)⋊1D6, C30.69(C2×D4), C2.28(D5×D12), (C10×D12)⋊16C2, C31(C23⋊D10), D6⋊Dic524C2, (C2×C60)⋊25C22, C10.28(C2×D12), C10.138(S3×D4), (C22×S3)⋊1D10, D10⋊C419S3, D303C422C2, (C6×Dic5)⋊3C22, (C22×D5).59D6, C2.29(C20⋊D6), (C2×C30).165C23, (C2×Dic15)⋊8C22, (C22×D15).57C22, (C2×C4)⋊2(S3×D5), (C22×S3×D5)⋊2C2, (C2×C5⋊D12)⋊9C2, (S3×C2×C10)⋊1C22, C2.19(S3×C5⋊D4), C6.41(C2×C5⋊D4), (C2×C15⋊D4)⋊10C2, (D5×C2×C6).42C22, C22.213(C2×S3×D5), (C3×D10⋊C4)⋊24C2, (C2×C6).177(C22×D5), (C2×C10).177(C22×S3), SmallGroup(480,551)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D304D4
C1C5C15C30C2×C30D5×C2×C6C22×S3×D5 — D304D4
C15C2×C30 — D304D4
C1C22C2×C4

Generators and relations for D304D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a15b, dbd=a25b, dcd=c-1 >

Subgroups: 1772 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×23], C5, S3 [×5], C6 [×3], C6 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], D5 [×4], C10 [×3], C10 [×3], Dic3, C12 [×2], D6 [×2], D6 [×17], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×2], C20, D10 [×2], D10 [×14], C2×C10, C2×C10 [×7], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3 [×2], C22×S3 [×7], C22×C6, C5×S3 [×3], C3×D5 [×2], D15 [×2], C30 [×3], C22≀C2, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×D5 [×7], C22×C10 [×2], D6⋊C4 [×2], C3×C22⋊C4, C2×D12, C2×D12, C2×C3⋊D4, S3×C23, C3×Dic5, Dic15, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×5], D30 [×2], D30 [×2], C2×C30, D10⋊C4, D10⋊C4, C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, D6⋊D4, C15⋊D4 [×2], C5⋊D12 [×2], C6×Dic5, C5×D12 [×2], C2×Dic15, C2×C60, C2×S3×D5 [×6], D5×C2×C6, S3×C2×C10 [×2], C22×D15, C23⋊D10, D6⋊Dic5, C3×D10⋊C4, D303C4, C2×C15⋊D4, C2×C5⋊D12, C10×D12, C22×S3×D5, D304D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, C22≀C2, C5⋊D4 [×2], C22×D5, C2×D12, S3×D4 [×2], S3×D5, D4×D5 [×2], C2×C5⋊D4, D6⋊D4, C2×S3×D5, C23⋊D10, D5×D12, C20⋊D6, S3×C5⋊D4, D304D4

Smallest permutation representation of D304D4
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 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(58 60)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)
(1 52 74 107)(2 53 75 108)(3 54 76 109)(4 55 77 110)(5 56 78 111)(6 57 79 112)(7 58 80 113)(8 59 81 114)(9 60 82 115)(10 31 83 116)(11 32 84 117)(12 33 85 118)(13 34 86 119)(14 35 87 120)(15 36 88 91)(16 37 89 92)(17 38 90 93)(18 39 61 94)(19 40 62 95)(20 41 63 96)(21 42 64 97)(22 43 65 98)(23 44 66 99)(24 45 67 100)(25 46 68 101)(26 47 69 102)(27 48 70 103)(28 49 71 104)(29 50 72 105)(30 51 73 106)
(1 107)(2 118)(3 99)(4 110)(5 91)(6 102)(7 113)(8 94)(9 105)(10 116)(11 97)(12 108)(13 119)(14 100)(15 111)(16 92)(17 103)(18 114)(19 95)(20 106)(21 117)(22 98)(23 109)(24 120)(25 101)(26 112)(27 93)(28 104)(29 115)(30 96)(31 83)(32 64)(33 75)(34 86)(35 67)(36 78)(37 89)(38 70)(39 81)(40 62)(41 73)(42 84)(43 65)(44 76)(45 87)(46 68)(47 79)(48 90)(49 71)(50 82)(51 63)(52 74)(53 85)(54 66)(55 77)(56 88)(57 69)(58 80)(59 61)(60 72)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E10A···10F10G···10N12A12B12C12D15A15B20A20B20C20D30A···30F60A···60H
order122222222223444556666610···1010···101212121215152020202030···3060···60
size11116610101230302420602222220202···212···124420204444444···44···4

60 irreducible representations

dim111111112222222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D5D6D6D6D10D10D12C5⋊D4S3×D4S3×D5D4×D5C2×S3×D5D5×D12C20⋊D6S3×C5⋊D4
kernelD304D4D6⋊Dic5C3×D10⋊C4D303C4C2×C15⋊D4C2×C5⋊D12C10×D12C22×S3×D5D10⋊C4C6×D5S3×C10D30C2×D12C2×Dic5C2×C20C22×D5C2×C12C22×S3D10D6C10C2×C4C6C22C2C2C2
# reps111111111222211124482242444

Matrix representation of D304D4 in GL4(𝔽61) generated by

06000
1100
00145
006017
,
06000
60000
00017
00180
,
234600
153800
001428
001747
,
234600
233800
001428
001747
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,1,60,0,0,45,17],[0,60,0,0,60,0,0,0,0,0,0,18,0,0,17,0],[23,15,0,0,46,38,0,0,0,0,14,17,0,0,28,47],[23,23,0,0,46,38,0,0,0,0,14,17,0,0,28,47] >;

D304D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_4D_4
% in TeX

G:=Group("D30:4D4");
// GroupNames label

G:=SmallGroup(480,551);
// by ID

G=gap.SmallGroup(480,551);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,100,1356,18822]);
// 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,d*a*d=a^11,c*b*c^-1=a^15*b,d*b*d=a^25*b,d*c*d=c^-1>;
// generators/relations

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