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G = Q83D30order 480 = 25·3·5

2nd semidirect product of Q8 and D30 acting via D30/D15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D30, C409D6, C249D10, Q83D30, D1209C2, D4.3D30, C1204C22, SD161D15, D30.31D4, D6015C22, C60.68C23, Dic15.36D4, (C5×Q8)⋊9D6, C55(Q83D6), (C3×Q8)⋊6D10, C40⋊S31C2, D4⋊D1511C2, (D4×D15)⋊10C2, C35(D40⋊C2), (C3×SD16)⋊1D5, (C5×SD16)⋊1S3, (C5×D4).15D6, C2.19(D4×D15), C6.112(D4×D5), Q83D158C2, C1531(C8⋊C22), (C15×SD16)⋊1C2, (C3×D4).15D10, C30.319(C2×D4), C10.114(S3×D4), C4.5(C22×D15), C153C816C22, Q82D1510C2, (Q8×C15)⋊13C22, C20.106(C22×S3), (D4×C15).22C22, (C4×D15).25C22, C12.106(C22×D5), SmallGroup(480,879)

Series: Derived Chief Lower central Upper central

C1C60 — Q83D30
C1C5C15C30C60C4×D15D4×D15 — Q83D30
C15C30C60 — Q83D30
C1C2C4SD16

Generators and relations for Q83D30
 G = < a,b,c,d | a4=c30=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 1092 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, C4×S3 [×2], D12 [×3], C3⋊D4, C3×D4, C3×Q8, C22×S3, D15 [×3], C30, C30, C8⋊C22, C52C8, C40, C4×D5 [×2], D20 [×3], C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, Dic15, C60, C60, D30, D30 [×4], C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, Q83D6, C153C8, C120, C4×D15, C4×D15, D60 [×2], D60, C157D4, D4×C15, Q8×C15, C22×D15, D40⋊C2, C40⋊S3, D120, D4⋊D15, Q82D15, C15×SD16, D4×D15, Q83D15, Q83D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C8⋊C22, C22×D5, S3×D4, D30 [×3], D4×D5, Q83D6, C22×D15, D40⋊C2, D4×D15, Q83D30

Smallest permutation representation of Q83D30
On 120 points
Generators in S120
(1 34 30 49)(2 50 16 35)(3 36 17 51)(4 52 18 37)(5 38 19 53)(6 54 20 39)(7 40 21 55)(8 56 22 41)(9 42 23 57)(10 58 24 43)(11 44 25 59)(12 60 26 45)(13 46 27 31)(14 32 28 47)(15 48 29 33)(61 76 119 104)(62 105 120 77)(63 78 91 106)(64 107 92 79)(65 80 93 108)(66 109 94 81)(67 82 95 110)(68 111 96 83)(69 84 97 112)(70 113 98 85)(71 86 99 114)(72 115 100 87)(73 88 101 116)(74 117 102 89)(75 90 103 118)
(1 97 30 69)(2 113 16 85)(3 99 17 71)(4 115 18 87)(5 101 19 73)(6 117 20 89)(7 103 21 75)(8 119 22 61)(9 105 23 77)(10 91 24 63)(11 107 25 79)(12 93 26 65)(13 109 27 81)(14 95 28 67)(15 111 29 83)(31 94 46 66)(32 82 47 110)(33 96 48 68)(34 84 49 112)(35 98 50 70)(36 86 51 114)(37 100 52 72)(38 88 53 116)(39 102 54 74)(40 90 55 118)(41 104 56 76)(42 62 57 120)(43 106 58 78)(44 64 59 92)(45 108 60 80)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D12A12B15A15B15C15D20A20B20C20D24A24B30A30B30C30D30E30F30G30H40A40B40C40D60A60B60C60D60E60F60G60H120A···120H
order1222223444556688101010101212151515152020202024243030303030303030404040406060606060606060120···120
size1143060602243022284602288482222448844222288884444444488884···4

60 irreducible representations

dim11111111222222222222224444444
type+++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10D15D30D30D30C8⋊C22S3×D4D4×D5Q83D6D40⋊C2D4×D15Q83D30
kernelQ83D30C40⋊S3D120D4⋊D15Q82D15C15×SD16D4×D15Q83D15C5×SD16Dic15D30C3×SD16C40C5×D4C5×Q8C24C3×D4C3×Q8SD16C8D4Q8C15C10C6C5C3C2C1
# reps11111111111211122244441122448

Matrix representation of Q83D30 in GL4(𝔽241) generated by

100103
01138138
2342342400
700240
,
0025206
0018160
21722700
21723100
,
6414818024
10917315156
0016193
0014884
,
30110238195
143211198226
0016193
0011080
G:=sub<GL(4,GF(241))| [1,0,234,7,0,1,234,0,0,138,240,0,103,138,0,240],[0,0,217,217,0,0,227,231,25,181,0,0,206,60,0,0],[64,109,0,0,148,173,0,0,180,15,161,148,24,156,93,84],[30,143,0,0,110,211,0,0,238,198,161,110,195,226,93,80] >;

Q83D30 in GAP, Magma, Sage, TeX

Q_8\rtimes_3D_{30}
% in TeX

G:=Group("Q8:3D30");
// GroupNames label

G:=SmallGroup(480,879);
// by ID

G=gap.SmallGroup(480,879);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,100,346,185,80,2693,18822]);
// Polycyclic

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

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