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G = D4.D30order 480 = 25·3·5

1st non-split extension by D4 of D30 acting via D30/C30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.6D30, C60.15D4, D6023C22, C60.75C23, Dic3020C22, (C6×D4)⋊2D5, (D4×C10)⋊2S3, (D4×C30)⋊2C2, (C2×D4)⋊2D15, D4⋊D1513C2, (C5×D4).30D6, (C2×C4).16D30, C1534(C8⋊C22), D4.D1513C2, (C3×D4).30D10, (C2×C30).146D4, C30.379(C2×D4), (C2×C20).145D6, C55(D126C22), C60.7C418C2, C153C817C22, (C2×C12).144D10, C35(D4.D10), C20.40(C3⋊D4), C12.42(C5⋊D4), C4.16(C157D4), (C2×C60).71C22, D6011C213C2, C4.12(C22×D15), C20.113(C22×S3), (D4×C15).35C22, C12.113(C22×D5), C22.10(C157D4), C2.9(C2×C157D4), C6.104(C2×C5⋊D4), (C2×C6).78(C5⋊D4), C10.104(C2×C3⋊D4), (C2×C10).78(C3⋊D4), SmallGroup(480,897)

Series: Derived Chief Lower central Upper central

C1C60 — D4.D30
C1C5C15C30C60D60D6011C2 — D4.D30
C15C30C60 — D4.D30
C1C2C2×C4C2×D4

Generators and relations for D4.D30
 G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c29 >

Subgroups: 692 in 136 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, D5, C10, C10 [×3], Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C3×D4, C22×C6, D15, C30, C30 [×3], C8⋊C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C4○D12, C6×D4, Dic15, C60 [×2], D30, C2×C30, C2×C30 [×4], C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], C4○D20, D4×C10, D126C22, C153C8 [×2], Dic30, C4×D15, D60, C157D4, C2×C60, D4×C15 [×2], D4×C15, C22×C30, D4.D10, C60.7C4, D4⋊D15 [×2], D4.D15 [×2], D6011C2, D4×C30, 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, D126C22, C157D4 [×2], C22×D15, D4.D10, C2×C157D4, D4.D30

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

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D6E6F6G8A8B10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222234445566666668810···1010···101212151515152020202030···3030···3060···60
size11244602226022222444460602···24···444222244442···24···44···4

81 irreducible representations

dim111111222222222222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4C3⋊D4D15C5⋊D4C5⋊D4D30D30C157D4C157D4C8⋊C22D126C22D4.D10D4.D30
kernelD4.D30C60.7C4D4⋊D15D4.D15D6011C2D4×C30D4×C10C60C2×C30C6×D4C2×C20C5×D4C2×C12C3×D4C20C2×C10C2×D4C12C2×C6C2×C4D4C4C22C15C5C3C1
# reps112211111212242244448881248

Matrix representation of D4.D30 in GL4(𝔽241) generated by

0100
240000
000240
0010
,
0100
1000
002400
0001
,
01000
231000
000217
00240
,
000217
00240
01000
231000
G:=sub<GL(4,GF(241))| [0,240,0,0,1,0,0,0,0,0,0,1,0,0,240,0],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,1],[0,231,0,0,10,0,0,0,0,0,0,24,0,0,217,0],[0,0,0,231,0,0,10,0,0,24,0,0,217,0,0,0] >;

D4.D30 in GAP, Magma, Sage, TeX

D_4.D_{30}
% in TeX

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

G:=SmallGroup(480,897);
// by ID

G=gap.SmallGroup(480,897);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,675,185,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^30=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^29>;
// generators/relations

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