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G = Dic10⋊Dic3order 480 = 25·3·5

1st semidirect product of Dic10 and Dic3 acting via Dic3/C3=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic101Dic3, C1511C4≀C2, D42(C3⋊F5), (C3×D4)⋊2F5, (D4×C15)⋊2C4, C60.34(C2×C4), C33(D4⋊F5), (C5×D4)⋊2Dic3, (C6×D5).33D4, (C4×D5).28D6, C12.10(C2×F5), C12.F57C2, (C3×Dic10)⋊1C4, D42D5.2S3, C20.2(C2×Dic3), C51(Q83Dic3), D10.6(C3⋊D4), (C3×Dic5).82D4, C6.21(C22⋊F5), C30.21(C22⋊C4), (D5×C12).69C22, Dic5.37(C3⋊D4), C10.6(C6.D4), C2.7(D10.D6), (C4×C3⋊F5)⋊6C2, C4.2(C2×C3⋊F5), (C3×D42D5).2C2, SmallGroup(480,313)

Series: Derived Chief Lower central Upper central

C1C60 — Dic10⋊Dic3
C1C5C15C30C3×Dic5D5×C12C12.F5 — Dic10⋊Dic3
C15C30C60 — Dic10⋊Dic3
C1C2C4D4

Generators and relations for Dic10⋊Dic3
 G = < a,b,c,d | a20=c6=1, b2=a10, d2=c3, bab-1=a-1, cac-1=a9, dad-1=a13, cbc-1=a10b, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 460 in 88 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C4≀C2, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C3⋊F5, C6×D5, C2×C30, C4.F5, C4×F5, D42D5, Q83Dic3, C15⋊C8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, C2×C3⋊F5, D4⋊F5, C12.F5, C4×C3⋊F5, C3×D42D5, Dic10⋊Dic3
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, D4⋊F5, D10.D6, Dic10⋊Dic3

Smallest permutation representation of Dic10⋊Dic3
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 70 11 80)(2 69 12 79)(3 68 13 78)(4 67 14 77)(5 66 15 76)(6 65 16 75)(7 64 17 74)(8 63 18 73)(9 62 19 72)(10 61 20 71)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(81 109 91 119)(82 108 92 118)(83 107 93 117)(84 106 94 116)(85 105 95 115)(86 104 96 114)(87 103 97 113)(88 102 98 112)(89 101 99 111)(90 120 100 110)
(1 90 31)(2 99 32 10 91 40)(3 88 33 19 92 29)(4 97 34 8 93 38)(5 86 35 17 94 27)(6 95 36)(7 84 37 15 96 25)(9 82 39 13 98 23)(11 100 21)(12 89 22 20 81 30)(14 87 24 18 83 28)(16 85 26)(41 68 112 47 62 118)(42 77 113 56 63 107)(43 66 114 45 64 116)(44 75 115 54 65 105)(46 73 117 52 67 103)(48 71 119 50 69 101)(49 80 120 59 70 110)(51 78 102 57 72 108)(53 76 104 55 74 106)(58 61 109 60 79 111)
(1 11)(2 8 10 4)(3 5 19 17)(6 16)(7 13 15 9)(12 18 20 14)(21 90)(22 87 30 83)(23 84 39 96)(24 81 28 89)(25 98 37 82)(26 95)(27 92 35 88)(29 86 33 94)(31 100)(32 97 40 93)(34 91 38 99)(36 85)(41 109 47 111)(42 106 56 104)(43 103 45 117)(44 120 54 110)(46 114 52 116)(48 108 50 102)(49 105 59 115)(51 119 57 101)(53 113 55 107)(58 118 60 112)(61 62 79 68)(63 76 77 74)(64 73 66 67)(65 70 75 80)(69 78 71 72)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H 5 6A6B6C6D8A8B10A10B10C12A12B12C12D12E15A15B 20 30A30B30C30D30E30F60A60B
order1222344444444566668810101012121212121515203030303030306060
size1141022552030303030424420606048841010202044844888888

39 irreducible representations

dim111111222222222444444488
type+++++++-+-+++-
imageC1C2C2C2C4C4S3D4D4Dic3D6Dic3C3⋊D4C3⋊D4C4≀C2F5C2×F5C3⋊F5C22⋊F5Q83Dic3C2×C3⋊F5D10.D6D4⋊F5Dic10⋊Dic3
kernelDic10⋊Dic3C12.F5C4×C3⋊F5C3×D42D5C3×Dic10D4×C15D42D5C3×Dic5C6×D5Dic10C4×D5C5×D4Dic5D10C15C3×D4C12D4C6C5C4C2C3C1
# reps111122111111224112222412

Matrix representation of Dic10⋊Dic3 in GL6(𝔽241)

17700000
0640000
000100
000010
000001
00240240240240
,
0640000
6400000
001000
00240240240240
000001
000010
,
100000
02400000
0012601212
00115127127115
0012120126
000229114229
,
24000000
0640000
001000
000001
000100
00240240240240

G:=sub<GL(6,GF(241))| [177,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,64,0,0,0,0,64,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,240,0,1,0,0,0,240,1,0],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,126,115,12,0,0,0,0,127,12,229,0,0,12,127,0,114,0,0,12,115,126,229],[240,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

Dic10⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,313);
// by ID

G=gap.SmallGroup(480,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,2693,14118,4724]);
// 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^13,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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