Copied to
clipboard

G = D4⋊D30order 480 = 25·3·5

2nd semidirect product of D4 and D30 acting via D30/C30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44D30, Q85D30, C60.206D4, C60.80C23, D60.40C22, C4○D43D15, (C5×D4)⋊19D6, (C5×Q8)⋊20D6, (C2×C30).8D4, D4⋊D1514C2, (C2×D60)⋊13C2, (C3×D4)⋊19D10, C55(D4⋊D6), (C3×Q8)⋊17D10, (C2×C4).21D30, C35(D4⋊D10), C1537(C8⋊C22), (C2×C20).158D6, C30.391(C2×D4), C60.7C421C2, C153C818C22, Q82D1514C2, (C2×C12).156D10, (D4×C15)⋊21C22, C4.24(C157D4), (C2×C60).83C22, (Q8×C15)⋊19C22, C4.17(C22×D15), C20.103(C3⋊D4), C12.103(C5⋊D4), C20.118(C22×S3), C12.118(C22×D5), C22.5(C157D4), (C5×C4○D4)⋊5S3, (C3×C4○D4)⋊1D5, (C15×C4○D4)⋊1C2, C6.118(C2×C5⋊D4), C2.23(C2×C157D4), (C2×C6).20(C5⋊D4), C10.118(C2×C3⋊D4), (C2×C10).19(C3⋊D4), SmallGroup(480,914)

Series: Derived Chief Lower central Upper central

C1C60 — D4⋊D30
C1C5C15C30C60D60C2×D60 — D4⋊D30
C15C30C60 — D4⋊D30
C1C2C2×C4C4○D4

Generators and relations for D4⋊D30
 G = < a,b,c,d | a4=c30=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab, dcd=c-1 >

Subgroups: 932 in 136 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, D15 [×2], C30, C30 [×2], C8⋊C22, C52C8 [×2], D20 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C60 [×2], C60, D30 [×4], C2×C30, C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, D4⋊D6, C153C8 [×2], D60 [×2], D60, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C22×D15, D4⋊D10, C60.7C4, D4⋊D15 [×2], Q82D15 [×2], C2×D60, C15×C4○D4, D4⋊D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, D4⋊D6, C157D4 [×2], C22×D15, D4⋊D10, C2×C157D4, D4⋊D30

Smallest permutation representation of D4⋊D30
On 120 points
Generators in S120
(1 51 21 31)(2 52 22 32)(3 53 23 33)(4 54 24 34)(5 55 25 35)(6 56 26 36)(7 57 27 37)(8 58 28 38)(9 59 29 39)(10 60 30 40)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(61 116 76 101)(62 117 77 102)(63 118 78 103)(64 119 79 104)(65 120 80 105)(66 91 81 106)(67 92 82 107)(68 93 83 108)(69 94 84 109)(70 95 85 110)(71 96 86 111)(72 97 87 112)(73 98 88 113)(74 99 89 114)(75 100 90 115)
(1 86 21 71)(2 72 22 87)(3 88 23 73)(4 74 24 89)(5 90 25 75)(6 76 26 61)(7 62 27 77)(8 78 28 63)(9 64 29 79)(10 80 30 65)(11 66 16 81)(12 82 17 67)(13 68 18 83)(14 84 19 69)(15 70 20 85)(31 111 51 96)(32 97 52 112)(33 113 53 98)(34 99 54 114)(35 115 55 100)(36 101 56 116)(37 117 57 102)(38 103 58 118)(39 119 59 104)(40 105 60 120)(41 91 46 106)(42 107 47 92)(43 93 48 108)(44 109 49 94)(45 95 50 110)
(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 5)(2 4)(6 15)(7 14)(8 13)(9 12)(10 11)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)(61 95)(62 94)(63 93)(64 92)(65 91)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D8A8B10A10B10C···10H12A12B12C12D12E15A15B15C15D20A20B20C20D20E···20J30A30B30C30D30E···30P60A···60H60I···60T
order122222344455666688101010···101212121212151515152020202020···203030303030···3060···6060···60
size1124606022242224446060224···422444222222224···422224···42···24···4

81 irreducible representations

dim111111222222222222222222224444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C3⋊D4C3⋊D4D15C5⋊D4C5⋊D4D30D30D30C157D4C157D4C8⋊C22D4⋊D6D4⋊D10D4⋊D30
kernelD4⋊D30C60.7C4D4⋊D15Q82D15C2×D60C15×C4○D4C5×C4○D4C60C2×C30C3×C4○D4C2×C20C5×D4C5×Q8C2×C12C3×D4C3×Q8C20C2×C10C4○D4C12C2×C6C2×C4D4Q8C4C22C15C5C3C1
# reps112211111211122222444444881248

Matrix representation of D4⋊D30 in GL6(𝔽241)

100000
010000
0020015600
00854100
00004185
0000156200
,
24000000
02400000
00002400
00000240
001000
000100
,
12400000
100000
0018924000
001000
0000521
00002400
,
100000
12400000
0018924000
00525200
0000122200
0000122119

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,200,85,0,0,0,0,156,41,0,0,0,0,0,0,41,156,0,0,0,0,85,200],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,240,0,0],[1,1,0,0,0,0,240,0,0,0,0,0,0,0,189,1,0,0,0,0,240,0,0,0,0,0,0,0,52,240,0,0,0,0,1,0],[1,1,0,0,0,0,0,240,0,0,0,0,0,0,189,52,0,0,0,0,240,52,0,0,0,0,0,0,122,122,0,0,0,0,200,119] >;

D4⋊D30 in GAP, Magma, Sage, TeX

D_4\rtimes D_{30}
% in TeX

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

G:=SmallGroup(480,914);
// by ID

G=gap.SmallGroup(480,914);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,675,185,80,2693,18822]);
// Polycyclic

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

׿
×
𝔽