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G = D30.27D4order 480 = 25·3·5

11st non-split extension by D30 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.27D4, D6⋊C414D5, D610(C4×D5), (C2×C20)⋊16D6, C6.66(D4×D5), D1012(C4×S3), (C2×C12)⋊16D10, C10.53(S3×D4), D6⋊Dic523C2, (C2×C60)⋊14C22, (C2×Dic5)⋊11D6, D30.32(C2×C4), C30.161(C2×D4), D10⋊C414S3, D152(C22⋊C4), (C2×Dic3)⋊11D10, C2.5(C20⋊D6), C30.69(C22×C4), (C6×Dic5)⋊2C22, (C22×D5).58D6, C2.4(D10⋊D6), D10⋊Dic323C2, (C2×C30).163C23, (C10×Dic3)⋊2C22, (C22×S3).49D10, (C2×Dic15)⋊29C22, (C22×D15).111C22, (C2×S3×D5)⋊3C4, C52(S3×C22⋊C4), C31(D5×C22⋊C4), C6.37(C2×C4×D5), C2.39(C4×S3×D5), (C6×D5)⋊8(C2×C4), (C2×C4×D15)⋊12C2, (C2×C4)⋊13(S3×D5), C155(C2×C22⋊C4), C10.70(S3×C2×C4), (C5×D6⋊C4)⋊14C2, (S3×C10)⋊15(C2×C4), (C22×S3×D5).3C2, C22.72(C2×S3×D5), (D5×C2×C6).41C22, (C2×D30.C2)⋊12C2, (S3×C2×C10).41C22, (C3×D10⋊C4)⋊14C2, (C2×C6).175(C22×D5), (C2×C10).175(C22×S3), SmallGroup(480,549)

Series: Derived Chief Lower central Upper central

C1C30 — D30.27D4
C1C5C15C30C2×C30D5×C2×C6C22×S3×D5 — D30.27D4
C15C30 — D30.27D4
C1C22C2×C4

Generators and relations for D30.27D4
 G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a15c-1 >

Subgroups: 1676 in 264 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22, C22 [×22], C5, S3 [×6], C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], C23 [×11], D5 [×6], C10 [×3], C10 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×16], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×2], D10 [×16], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3 [×9], C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×4], C30 [×3], C2×C22⋊C4, C4×D5 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5 [×9], C22×C10, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×6], C2×C30, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, S3×C22⋊C4, D30.C2 [×2], C6×Dic5, C10×Dic3, C4×D15 [×2], C2×Dic15, C2×C60, C2×S3×D5 [×4], C2×S3×D5 [×4], D5×C2×C6, S3×C2×C10, C22×D15, D5×C22⋊C4, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C2×D30.C2, C2×C4×D15, C22×S3×D5, D30.27D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C2×C22⋊C4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4 [×2], S3×D5, C2×C4×D5, D4×D5 [×2], S3×C22⋊C4, C2×S3×D5, D5×C22⋊C4, C4×S3×D5, C20⋊D6, D10⋊D6, D30.27D4

Smallest permutation representation of D30.27D4
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 36)(32 35)(33 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(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)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 116 72 34)(2 117 73 35)(3 118 74 36)(4 119 75 37)(5 120 76 38)(6 91 77 39)(7 92 78 40)(8 93 79 41)(9 94 80 42)(10 95 81 43)(11 96 82 44)(12 97 83 45)(13 98 84 46)(14 99 85 47)(15 100 86 48)(16 101 87 49)(17 102 88 50)(18 103 89 51)(19 104 90 52)(20 105 61 53)(21 106 62 54)(22 107 63 55)(23 108 64 56)(24 109 65 57)(25 110 66 58)(26 111 67 59)(27 112 68 60)(28 113 69 31)(29 114 70 32)(30 115 71 33)
(1 49 16 34)(2 60 17 45)(3 41 18 56)(4 52 19 37)(5 33 20 48)(6 44 21 59)(7 55 22 40)(8 36 23 51)(9 47 24 32)(10 58 25 43)(11 39 26 54)(12 50 27 35)(13 31 28 46)(14 42 29 57)(15 53 30 38)(61 100 76 115)(62 111 77 96)(63 92 78 107)(64 103 79 118)(65 114 80 99)(66 95 81 110)(67 106 82 91)(68 117 83 102)(69 98 84 113)(70 109 85 94)(71 120 86 105)(72 101 87 116)(73 112 88 97)(74 93 89 108)(75 104 90 119)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222222222344444444556666610···1010101010121212121515202020202020202030···3060···60
size11116610101515151522266101030302222220202···212121212442020444444121212124···44···4

66 irreducible representations

dim111111111222222222224444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D4D5D6D6D6D10D10D10C4×S3C4×D5S3×D4S3×D5D4×D5C2×S3×D5C4×S3×D5C20⋊D6D10⋊D6
kernelD30.27D4D10⋊Dic3D6⋊Dic5C3×D10⋊C4C5×D6⋊C4C2×D30.C2C2×C4×D15C22×S3×D5C2×S3×D5D10⋊C4D30D6⋊C4C2×Dic5C2×C20C22×D5C2×Dic3C2×C12C22×S3D10D6C10C2×C4C6C22C2C2C2
# reps111111118142111222482242444

Matrix representation of D30.27D4 in GL6(𝔽61)

60150000
1220000
00606000
00191800
0000600
0000060
,
60150000
010000
0001700
0018000
0000600
0000060
,
5000000
0500000
0060000
0006000
0000120
0000060
,
5000000
10110000
0060000
0006000
0000120
0000660

G:=sub<GL(6,GF(61))| [60,12,0,0,0,0,15,2,0,0,0,0,0,0,60,19,0,0,0,0,60,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,15,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,20,60],[50,10,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,6,0,0,0,0,20,60] >;

D30.27D4 in GAP, Magma, Sage, TeX

D_{30}._{27}D_4
% in TeX

G:=Group("D30.27D4");
// GroupNames label

G:=SmallGroup(480,549);
// by ID

G=gap.SmallGroup(480,549);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

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