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G = D308D4order 480 = 25·3·5

8th semidirect product of D30 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D308D4, (C6×D5)⋊8D4, (S3×C10)⋊8D4, C232(S3×D5), C1510C22≀C2, C53(C232D6), D66(C5⋊D4), (C2×Dic5)⋊7D6, C6.167(D4×D5), (C22×C6)⋊3D10, (C22×C10)⋊6D6, D106(C3⋊D4), C33(C23⋊D10), D6⋊Dic538C2, (C2×Dic3)⋊7D10, C10.167(S3×D4), C30.257(C2×D4), D304C438C2, (C22×C30)⋊6C22, (C22×D5).66D6, D10⋊Dic338C2, (C2×C30).219C23, (C6×Dic5)⋊11C22, (C22×S3).57D10, C2.47(D10⋊D6), (C2×Dic15)⋊15C22, (C10×Dic3)⋊11C22, (C22×D15).71C22, (C22×S3×D5)⋊4C2, (C2×C5⋊D4)⋊10S3, (C2×C3⋊D4)⋊10D5, (C6×C5⋊D4)⋊10C2, C2.47(D5×C3⋊D4), C6.70(C2×C5⋊D4), C2.47(S3×C5⋊D4), (C10×C3⋊D4)⋊10C2, (C2×C157D4)⋊21C2, C10.71(C2×C3⋊D4), (D5×C2×C6).57C22, C22.248(C2×S3×D5), (S3×C2×C10).57C22, (C2×C6).231(C22×D5), (C2×C10).231(C22×S3), SmallGroup(480,653)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D308D4
C1C5C15C30C2×C30D5×C2×C6C22×S3×D5 — D308D4
C15C2×C30 — D308D4
C1C22C23

Generators and relations for D308D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=a25b, dbd=a10b, dcd=c-1 >

Subgroups: 1692 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×23], C5, S3 [×4], C6 [×3], C6 [×3], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5 [×4], C10 [×3], C10 [×3], Dic3 [×2], C12, D6 [×2], D6 [×14], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×2], C20, D10 [×2], D10 [×14], C2×C10, C2×C10 [×7], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×S3 [×7], C22×C6, C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30 [×3], C30, C22≀C2, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×D5 [×7], C22×C10, C22×C10, D6⋊C4 [×2], C6.D4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C3×Dic5, Dic15, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×3], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23×D5, C232D6, C6×Dic5, C3×C5⋊D4 [×2], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15, C157D4 [×2], C2×S3×D5 [×6], D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, C23⋊D10, D10⋊Dic3, D6⋊Dic5, D304C4, C6×C5⋊D4, C10×C3⋊D4, C2×C157D4, C22×S3×D5, D308D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C5⋊D4 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, D4×D5 [×2], C2×C5⋊D4, C232D6, C2×S3×D5, C23⋊D10, D5×C3⋊D4, S3×C5⋊D4, D10⋊D6, D308D4

Smallest permutation representation of D308D4
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 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(83 90)(84 89)(85 88)(86 87)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(114 120)(115 119)(116 118)
(1 95 72 46)(2 106 73 57)(3 117 74 38)(4 98 75 49)(5 109 76 60)(6 120 77 41)(7 101 78 52)(8 112 79 33)(9 93 80 44)(10 104 81 55)(11 115 82 36)(12 96 83 47)(13 107 84 58)(14 118 85 39)(15 99 86 50)(16 110 87 31)(17 91 88 42)(18 102 89 53)(19 113 90 34)(20 94 61 45)(21 105 62 56)(22 116 63 37)(23 97 64 48)(24 108 65 59)(25 119 66 40)(26 100 67 51)(27 111 68 32)(28 92 69 43)(29 103 70 54)(30 114 71 35)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 110)(32 91)(33 102)(34 113)(35 94)(36 105)(37 116)(38 97)(39 108)(40 119)(41 100)(42 111)(43 92)(44 103)(45 114)(46 95)(47 106)(48 117)(49 98)(50 109)(51 120)(52 101)(53 112)(54 93)(55 104)(56 115)(57 96)(58 107)(59 118)(60 99)(61 71)(62 82)(64 74)(65 85)(67 77)(68 88)(70 80)(73 83)(76 86)(79 89)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A···30N
order12222222222344455666666610···101010101010101010121215152020202030···30
size1111466101030302122060222224420202···2444412121212202044121212124···4

60 irreducible representations

dim1111111122222222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D5D6D6D6D10D10D10C3⋊D4C5⋊D4S3×D4S3×D5D4×D5C2×S3×D5D5×C3⋊D4S3×C5⋊D4D10⋊D6
kernelD308D4D10⋊Dic3D6⋊Dic5D304C4C6×C5⋊D4C10×C3⋊D4C2×C157D4C22×S3×D5C2×C5⋊D4C6×D5S3×C10D30C2×C3⋊D4C2×Dic5C22×D5C22×C10C2×Dic3C22×S3C22×C6D10D6C10C23C6C22C2C2C2
# reps1111111112222111222482242444

Matrix representation of D308D4 in GL6(𝔽61)

0600000
110000
00176000
001000
0000600
0000060
,
0600000
6000000
00176000
00444400
0000600
0000311
,
9180000
9520000
0060000
0006000
0000475
00002214
,
100000
60600000
001000
000100
000010
000001

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,17,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,17,44,0,0,0,0,60,44,0,0,0,0,0,0,60,31,0,0,0,0,0,1],[9,9,0,0,0,0,18,52,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,47,22,0,0,0,0,5,14],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D308D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_8D_4
% in TeX

G:=Group("D30:8D4");
// GroupNames label

G:=SmallGroup(480,653);
// by ID

G=gap.SmallGroup(480,653);
# by ID

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

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=a^25*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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