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G = D202Dic3order 480 = 25·3·5

2nd semidirect product of D20 and Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D202Dic3, C1513C4≀C2, Q83(C3⋊F5), (C3×Q8)⋊2F5, (C3×D20)⋊2C4, (Q8×C15)⋊2C4, C60.36(C2×C4), (C5×Q8)⋊5Dic3, (C6×D5).34D4, C33(Q82F5), (C4×D5).30D6, C12.12(C2×F5), C12.F58C2, Q82D5.4S3, C20.4(C2×Dic3), C52(Q83Dic3), D10.7(C3⋊D4), (C3×Dic5).83D4, C6.23(C22⋊F5), C30.23(C22⋊C4), (D5×C12).71C22, Dic5.38(C3⋊D4), C10.8(C6.D4), C2.9(D10.D6), (C4×C3⋊F5)⋊7C2, C4.4(C2×C3⋊F5), (C3×Q82D5).2C2, SmallGroup(480,315)

Series: Derived Chief Lower central Upper central

C1C60 — D202Dic3
C1C5C15C30C3×Dic5D5×C12C12.F5 — D202Dic3
C15C30C60 — D202Dic3
C1C2C4Q8

Generators and relations for D202Dic3
 G = < a,b,c,d | a20=b2=c6=1, d2=c3, bab=a-1, cac-1=a9, dad-1=a13, cbc-1=a18b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 492 in 88 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, F5, D10, D10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C4≀C2, C5⋊C8, C4×D5, C4×D5, D20, D20, C5×Q8, C2×F5, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, C60, C60, C3⋊F5, C6×D5, C6×D5, C4.F5, C4×F5, Q82D5, Q83Dic3, C15⋊C8, D5×C12, D5×C12, C3×D20, C3×D20, Q8×C15, C2×C3⋊F5, Q82F5, C12.F5, C4×C3⋊F5, C3×Q82D5, D202Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C4≀C2, C2×F5, C6.D4, C3⋊F5, C22⋊F5, Q83Dic3, C2×C3⋊F5, Q82F5, D10.D6, D202Dic3

Smallest permutation representation of D202Dic3
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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 100)(57 99)(58 98)(59 97)(60 96)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)
(1 81 61 11 91 71)(2 90 62 20 92 80)(3 99 63 9 93 69)(4 88 64 18 94 78)(5 97 65 7 95 67)(6 86 66 16 96 76)(8 84 68 14 98 74)(10 82 70 12 100 72)(13 89 73 19 83 79)(15 87 75 17 85 77)(21 53 114 25 49 118)(22 42 115 34 50 107)(23 51 116)(24 60 117 32 52 105)(26 58 119 30 54 103)(27 47 120 39 55 112)(28 56 101)(29 45 102 37 57 110)(31 43 104 35 59 108)(33 41 106)(36 48 109 40 44 113)(38 46 111)
(1 16 11 6)(2 13 20 19)(3 10 9 12)(4 7 18 5)(8 15 14 17)(21 29 25 37)(22 26 34 30)(24 40 32 36)(27 31 39 35)(41 106)(42 103 50 119)(43 120 59 112)(44 117 48 105)(45 114 57 118)(46 111)(47 108 55 104)(49 102 53 110)(51 116)(52 113 60 109)(54 107 58 115)(56 101)(61 86 71 96)(62 83 80 89)(63 100 69 82)(64 97 78 95)(65 94 67 88)(66 91 76 81)(68 85 74 87)(70 99 72 93)(73 90 79 92)(75 84 77 98)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H 5 6A6B6C6D8A8B 10 12A12B12C12D12E15A15B20A20B20C30A30B60A···60F
order122234444444456666881012121212121515202020303060···60
size11102022455303030304220202060604444101044888448···8

39 irreducible representations

dim111111222222222444444488
type++++++++--++++
imageC1C2C2C2C4C4S3D4D4D6Dic3Dic3C3⋊D4C3⋊D4C4≀C2F5C2×F5C3⋊F5C22⋊F5Q83Dic3C2×C3⋊F5D10.D6Q82F5D202Dic3
kernelD202Dic3C12.F5C4×C3⋊F5C3×Q82D5C3×D20Q8×C15Q82D5C3×Dic5C6×D5C4×D5D20C5×Q8Dic5D10C15C3×Q8C12Q8C6C5C4C2C3C1
# reps111122111111224112222412

Matrix representation of D202Dic3 in GL8(𝔽241)

640000000
064000000
103017700000
396401770000
0000240100
0000240010
0000240001
0000240000
,
850200000
86240020000
3015600000
3015510000
00001000
0000100240
0000102400
0000124000
,
0240000000
11000000
4342010000
156442402400000
0000229012114
0000229121260
0000012612229
0000114120229
,
1770000000
6464000000
2320100000
1681532402400000
0000229121270
0000115120229
0000229012115
0000012712229

G:=sub<GL(8,GF(241))| [64,0,103,39,0,0,0,0,0,64,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[85,86,3,3,0,0,0,0,0,240,0,0,0,0,0,0,2,0,156,155,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0],[0,1,43,156,0,0,0,0,240,1,42,44,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,229,229,0,114,0,0,0,0,0,12,126,12,0,0,0,0,12,126,12,0,0,0,0,0,114,0,229,229],[177,64,232,168,0,0,0,0,0,64,0,153,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,229,115,229,0,0,0,0,0,12,12,0,127,0,0,0,0,127,0,12,12,0,0,0,0,0,229,115,229] >;

D202Dic3 in GAP, Magma, Sage, TeX

D_{20}\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("D20:2Dic3");
// GroupNames label

G:=SmallGroup(480,315);
// by ID

G=gap.SmallGroup(480,315);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,100,675,346,80,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^13,c*b*c^-1=a^18*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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