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G = Dic102Dic3order 480 = 25·3·5

2nd semidirect product of Dic10 and Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic102Dic3, Q82(C3⋊F5), (C3×Q8)⋊1F5, (Q8×C15)⋊1C4, C5⋊(Q82Dic3), (Q8×D5).4S3, C60.35(C2×C4), (C5×Q8)⋊4Dic3, C33(Q8⋊F5), (C4×D5).29D6, (C6×D5).77D4, C12.11(C2×F5), (C3×D5).6Q16, C60⋊C4.5C2, (C3×Dic10)⋊2C4, C159(Q8⋊C4), C20.3(C2×Dic3), C60.C4.4C2, (C3×Dic5).38D4, (C3×D5).10SD16, D5.3(C3⋊Q16), C6.22(C22⋊F5), D10.34(C3⋊D4), C30.22(C22⋊C4), D5.3(Q82S3), Dic5.7(C3⋊D4), (D5×C12).70C22, C10.7(C6.D4), C2.8(D10.D6), C4.3(C2×C3⋊F5), (C3×Q8×D5).2C2, SmallGroup(480,314)

Series: Derived Chief Lower central Upper central

C1C60 — Dic102Dic3
C1C5C15C30C6×D5D5×C12C60⋊C4 — Dic102Dic3
C15C30C60 — Dic102Dic3
C1C2C4Q8

Generators and relations for Dic102Dic3
 G = < a,b,c,d | a20=c6=1, b2=a10, d2=c3, bab-1=a-1, cac-1=a9, dad-1=a7, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 444 in 84 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22, C5, C6, C6 [×2], C8, C2×C4 [×3], Q8, Q8 [×2], D5 [×2], C10, Dic3, C12, C12 [×3], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C3⋊C8, C2×Dic3, C2×C12 [×2], C3×Q8, C3×Q8 [×2], C3×D5 [×2], C30, Q8⋊C4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C3⋊F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, Q82Dic3, C15⋊C8, C3×Dic10, C3×Dic10, D5×C12, D5×C12, Q8×C15, C2×C3⋊F5, Q8⋊F5, C60.C4, C60⋊C4, C3×Q8×D5, Dic102Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, SD16, Q16, F5, C2×Dic3, C3⋊D4 [×2], Q8⋊C4, C2×F5, Q82S3, C3⋊Q16, C6.D4, C3⋊F5, C22⋊F5, Q82Dic3, C2×C3⋊F5, Q8⋊F5, D10.D6, Dic102Dic3

Smallest permutation representation of Dic102Dic3
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 11 27)(2 36 12 26)(3 35 13 25)(4 34 14 24)(5 33 15 23)(6 32 16 22)(7 31 17 21)(8 30 18 40)(9 29 19 39)(10 28 20 38)(41 86 51 96)(42 85 52 95)(43 84 53 94)(44 83 54 93)(45 82 55 92)(46 81 56 91)(47 100 57 90)(48 99 58 89)(49 98 59 88)(50 97 60 87)(61 118 71 108)(62 117 72 107)(63 116 73 106)(64 115 74 105)(65 114 75 104)(66 113 76 103)(67 112 77 102)(68 111 78 101)(69 110 79 120)(70 109 80 119)
(1 67 51)(2 76 52 10 68 60)(3 65 53 19 69 49)(4 74 54 8 70 58)(5 63 55 17 71 47)(6 72 56)(7 61 57 15 73 45)(9 79 59 13 75 43)(11 77 41)(12 66 42 20 78 50)(14 64 44 18 80 48)(16 62 46)(21 108 100 33 116 92)(22 117 81)(23 106 82 31 118 90)(24 115 83 40 119 99)(25 104 84 29 120 88)(26 113 85 38 101 97)(27 102 86)(28 111 87 36 103 95)(30 109 89 34 105 93)(32 107 91)(35 114 94 39 110 98)(37 112 96)
(2 4 10 8)(3 7 19 15)(5 13 17 9)(6 16)(12 14 20 18)(21 34 33 30)(22 37)(23 40 31 24)(25 26 29 38)(27 32)(28 35 36 39)(41 77)(42 80 50 64)(43 63 59 71)(44 66 48 78)(45 69 57 65)(46 72)(47 75 55 79)(49 61 53 73)(51 67)(52 70 60 74)(54 76 58 68)(56 62)(81 112)(82 115 90 119)(83 118 99 106)(84 101 88 113)(85 104 97 120)(86 107)(87 110 95 114)(89 116 93 108)(91 102)(92 105 100 109)(94 111 98 103)(96 117)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D 10 12A12B12C12D12E12F15A15B20A20B20C30A30B60A···60F
order1222344444456668888101212121212121515202020303060···60
size11552241020606042101030303030444420202044888448···8

39 irreducible representations

dim11111122222222224444444488
type+++++++-+--+++-+-
imageC1C2C2C2C4C4S3D4D4Dic3D6Dic3SD16Q16C3⋊D4C3⋊D4F5C2×F5Q82S3C3⋊Q16C3⋊F5C22⋊F5C2×C3⋊F5D10.D6Q8⋊F5Dic102Dic3
kernelDic102Dic3C60.C4C60⋊C4C3×Q8×D5C3×Dic10Q8×C15Q8×D5C3×Dic5C6×D5Dic10C4×D5C5×Q8C3×D5C3×D5Dic5D10C3×Q8C12D5D5Q8C6C4C2C3C1
# reps11112211111122221111222412

Matrix representation of Dic102Dic3 in GL6(𝔽241)

24030000
16010000
000100
000010
000001
00240240240240
,
01840000
14800000
00000240
00002400
00024000
00240000
,
24000000
02400000
0012120126
000229114229
002291142290
0012601212
,
17700000
118640000
000100
000001
001000
000010

G:=sub<GL(6,GF(241))| [240,160,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,148,0,0,0,0,184,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,12,0,229,126,0,0,12,229,114,0,0,0,0,114,229,12,0,0,126,229,0,12],[177,118,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0] >;

Dic102Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("Dic10:2Dic3");
// GroupNames label

G:=SmallGroup(480,314);
// by ID

G=gap.SmallGroup(480,314);
# by ID

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

G:=Group<a,b,c,d|a^20=c^6=1,b^2=a^10,d^2=c^3,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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