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G = D305D4order 480 = 25·3·5

5th semidirect product of D30 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D305D4, D65D20, D105D12, D6⋊C48D5, (C6×D5)⋊5D4, (C2×C20)⋊2D6, C153C22≀C2, (C2×D60)⋊3C2, (S3×C10)⋊5D4, (C2×C12)⋊2D10, C6.27(D4×D5), C51(D6⋊D4), (C2×C60)⋊1C22, (C2×Dic5)⋊2D6, C6.28(C2×D20), C2.29(D5×D12), C30.70(C2×D4), C2.29(S3×D20), C10.27(S3×D4), D10⋊C48S3, C31(C22⋊D20), (C2×Dic3)⋊2D10, C10.29(C2×D12), D304C424C2, (C6×Dic5)⋊4C22, (C22×D5).60D6, (C2×C30).166C23, (C10×Dic3)⋊4C22, (C22×S3).51D10, (C22×D15)⋊5C22, C2.19(D10⋊D6), (C2×C4)⋊3(S3×D5), (C5×D6⋊C4)⋊8C2, (C22×S3×D5)⋊3C2, (C2×C3⋊D20)⋊10C2, (C2×C5⋊D12)⋊10C2, (C3×D10⋊C4)⋊8C2, (D5×C2×C6).43C22, C22.214(C2×S3×D5), (S3×C2×C10).43C22, (C2×C6).178(C22×D5), (C2×C10).178(C22×S3), SmallGroup(480,552)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D305D4
C1C5C15C30C2×C30D5×C2×C6C22×S3×D5 — D305D4
C15C2×C30 — D305D4
C1C22C2×C4

Generators and relations for D305D4
 G = < a,b,c,d | a6=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 1964 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 [×5], C10 [×3], C10 [×2], Dic3, C12 [×2], D6 [×2], D6 [×17], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×2], D10 [×17], C2×C10, C2×C10 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×S3 [×8], C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×3], C30 [×3], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×D5 [×8], C22×C10, D6⋊C4, D6⋊C4, C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, C5×Dic3, C3×Dic5, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×5], C2×C30, D10⋊C4, D10⋊C4, C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, D6⋊D4, C3⋊D20 [×2], C5⋊D12 [×2], C6×Dic5, C10×Dic3, D60 [×2], C2×C60, C2×S3×D5 [×6], D5×C2×C6, S3×C2×C10, C22×D15 [×2], C22⋊D20, D304C4, C3×D10⋊C4, C5×D6⋊C4, C2×C3⋊D20, C2×C5⋊D12, C2×D60, C22×S3×D5, D305D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, C22≀C2, D20 [×2], C22×D5, C2×D12, S3×D4 [×2], S3×D5, C2×D20, D4×D5 [×2], D6⋊D4, C2×S3×D5, C22⋊D20, D5×D12, S3×D20, D10⋊D6, D305D4

Smallest permutation representation of D305D4
On 120 points
Generators in S120
(1 84 101 21 58 76)(2 85 102 22 59 77)(3 86 103 23 60 78)(4 87 104 24 41 79)(5 88 105 25 42 80)(6 89 106 26 43 61)(7 90 107 27 44 62)(8 91 108 28 45 63)(9 92 109 29 46 64)(10 93 110 30 47 65)(11 94 111 31 48 66)(12 95 112 32 49 67)(13 96 113 33 50 68)(14 97 114 34 51 69)(15 98 115 35 52 70)(16 99 116 36 53 71)(17 100 117 37 54 72)(18 81 118 38 55 73)(19 82 119 39 56 74)(20 83 120 40 57 75)
(1 111)(2 67)(3 113)(4 69)(5 115)(6 71)(7 117)(8 73)(9 119)(10 75)(11 101)(12 77)(13 103)(14 79)(15 105)(16 61)(17 107)(18 63)(19 109)(20 65)(21 66)(22 112)(23 68)(24 114)(25 70)(26 116)(27 72)(28 118)(29 74)(30 120)(31 76)(32 102)(33 78)(34 104)(35 80)(36 106)(37 62)(38 108)(39 64)(40 110)(41 97)(42 52)(43 99)(44 54)(45 81)(46 56)(47 83)(48 58)(49 85)(50 60)(51 87)(53 89)(55 91)(57 93)(59 95)(82 92)(84 94)(86 96)(88 98)(90 100)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 107)(42 106)(43 105)(44 104)(45 103)(46 102)(47 101)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 108)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222222223444556666610···1010101010121212121515202020202020202030···3060···60
size11116610103030602412202222220202···212121212442020444444121212124···44···4

60 irreducible representations

dim1111111122222222222224444444
type++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D5D6D6D6D10D10D10D12D20S3×D4S3×D5D4×D5C2×S3×D5D5×D12S3×D20D10⋊D6
kernelD305D4D304C4C3×D10⋊C4C5×D6⋊C4C2×C3⋊D20C2×C5⋊D12C2×D60C22×S3×D5D10⋊C4C6×D5S3×C10D30D6⋊C4C2×Dic5C2×C20C22×D5C2×Dic3C2×C12C22×S3D10D6C10C2×C4C6C22C2C2C2
# reps1111111112222111222482242444

Matrix representation of D305D4 in GL6(𝔽61)

100000
010000
00606000
001000
0000600
0000060
,
6000000
0600000
00606000
000100
0000600
0000481
,
7320000
2920000
0060000
0006000
00005320
0000128
,
7320000
29540000
0060000
001100
0000841
00005553

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,60,48,0,0,0,0,0,1],[7,29,0,0,0,0,32,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,53,12,0,0,0,0,20,8],[7,29,0,0,0,0,32,54,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,8,55,0,0,0,0,41,53] >;

D305D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_5D_4
% in TeX

G:=Group("D30:5D4");
// GroupNames label

G:=SmallGroup(480,552);
// by ID

G=gap.SmallGroup(480,552);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^20=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

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