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G = Dic5.4Dic6order 480 = 25·3·5

4th non-split extension by Dic5 of Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.4Dic6, C3⋊C82F5, C15⋊(C2.D8), C5⋊(C6.Q16), C153C82C4, C4⋊F5.4S3, C32(D5.D8), (C3×D5).3D8, C6.9(C4⋊F5), C4.16(S3×F5), C12.6(C2×F5), C30.2(C4⋊C4), C20.16(C4×S3), C60.16(C2×C4), (C6×D5).26D4, (C4×D5).61D6, (C3×D5).3Q16, C60⋊C4.4C2, D5.1(D4⋊S3), (C3×Dic5).4Q8, C2.5(Dic3⋊F5), D5.1(C3⋊Q16), D10.14(C3⋊D4), C10.2(Dic3⋊C4), (D5×C12).47C22, (C5×C3⋊C8)⋊2C4, (D5×C3⋊C8).3C2, (C3×C4⋊F5).4C2, SmallGroup(480,236)

Series: Derived Chief Lower central Upper central

C1C60 — Dic5.4Dic6
C1C5C15C30C6×D5D5×C12C3×C4⋊F5 — Dic5.4Dic6
C15C30C60 — Dic5.4Dic6
C1C2C4

Generators and relations for Dic5.4Dic6
 G = < a,b,c,d | a10=c12=1, b2=a5, d2=a5bc6, bab-1=a-1, cac-1=a3, ad=da, cbc-1=a5b, bd=db, dcd-1=a5bc-1 >

Subgroups: 404 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×3], C22, C5, C6, C6 [×2], C8 [×2], C2×C4 [×3], D5 [×2], C10, Dic3, C12, C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5, C20, F5 [×2], D10, C3⋊C8, C3⋊C8, C2×Dic3, C2×C12 [×2], C3×D5 [×2], C30, C2.D8, C52C8, C40, C4×D5, C2×F5 [×2], C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, C60, C3×F5, C3⋊F5, C6×D5, C8×D5, C4⋊F5, C4⋊F5, C6.Q16, C5×C3⋊C8, C153C8, D5×C12, C6×F5, C2×C3⋊F5, D5.D8, D5×C3⋊C8, C3×C4⋊F5, C60⋊C4, Dic5.4Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D8, Q16, F5, Dic6, C4×S3, C3⋊D4, C2.D8, C2×F5, Dic3⋊C4, D4⋊S3, C3⋊Q16, C4⋊F5, C6.Q16, S3×F5, D5.D8, Dic3⋊F5, Dic5.4Dic6

Smallest permutation representation of Dic5.4Dic6
On 120 points
Generators in S120
(1 43 60 120 64 12 70 114 54 37)(2 109 71 38 49 7 55 44 65 115)(3 39 56 116 72 8 66 110 50 45)(4 117 67 46 57 9 51 40 61 111)(5 47 52 112 68 10 62 118 58 41)(6 113 63 42 53 11 59 48 69 119)(13 97 89 80 26 20 32 74 95 103)(14 81 33 104 90 21 96 98 27 75)(15 105 85 76 34 22 28 82 91 99)(16 77 29 100 86 23 92 106 35 83)(17 101 93 84 30 24 36 78 87 107)(18 73 25 108 94 19 88 102 31 79)
(1 20 12 13)(2 14 7 21)(3 22 8 15)(4 16 9 23)(5 24 10 17)(6 18 11 19)(25 42 102 69)(26 70 103 43)(27 44 104 71)(28 72 105 45)(29 46 106 61)(30 62 107 47)(31 48 108 63)(32 64 97 37)(33 38 98 65)(34 66 99 39)(35 40 100 67)(36 68 101 41)(49 96 115 81)(50 82 116 85)(51 86 117 83)(52 84 118 87)(53 88 119 73)(54 74 120 89)(55 90 109 75)(56 76 110 91)(57 92 111 77)(58 78 112 93)(59 94 113 79)(60 80 114 95)
(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 19 13 11 12 18 20 6)(2 5 21 17 7 10 14 24)(3 23 15 9 8 16 22 4)(25 74 63 60 102 89 48 114)(26 119 37 94 103 53 64 79)(27 84 65 58 104 87 38 112)(28 117 39 92 105 51 66 77)(29 82 67 56 106 85 40 110)(30 115 41 90 107 49 68 75)(31 80 69 54 108 95 42 120)(32 113 43 88 97 59 70 73)(33 78 71 52 98 93 44 118)(34 111 45 86 99 57 72 83)(35 76 61 50 100 91 46 116)(36 109 47 96 101 55 62 81)

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

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

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

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D 10 12A12B···12F 15 20A20B 30 40A40B40C40D60A60B
order1222344444456668888101212···1215202030404040406060
size11552210202060604210106630304420···2084481212121288

36 irreducible representations

dim111111222222222444444888
type+++++-+++--+++-+-
imageC1C2C2C2C4C4S3Q8D4D6D8Q16Dic6C4×S3C3⋊D4F5C2×F5D4⋊S3C3⋊Q16C4⋊F5D5.D8S3×F5Dic3⋊F5Dic5.4Dic6
kernelDic5.4Dic6D5×C3⋊C8C3×C4⋊F5C60⋊C4C5×C3⋊C8C153C8C4⋊F5C3×Dic5C6×D5C4×D5C3×D5C3×D5Dic5C20D10C3⋊C8C12D5D5C6C3C4C2C1
# reps111122111122222111124112

Matrix representation of Dic5.4Dic6 in GL8(𝔽241)

2400000000
0240000000
0024000000
0002400000
00000100
00000010
00000001
0000240240240240
,
00100000
00010000
2400000000
0240000000
0000000240
0000002400
0000024000
0000240000
,
1012231620000
2012101952200000
231622312290000
19522040310000
00000341734
00002240207207
00002072070224
00003417340
,
2201792102290000
862186310000
31122201790000
15521086210000
00002240207207
00003417340
00000341734
00002072070224

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0],[10,201,231,195,0,0,0,0,12,210,62,220,0,0,0,0,231,195,231,40,0,0,0,0,62,220,229,31,0,0,0,0,0,0,0,0,0,224,207,34,0,0,0,0,34,0,207,17,0,0,0,0,17,207,0,34,0,0,0,0,34,207,224,0],[220,86,31,155,0,0,0,0,179,21,12,210,0,0,0,0,210,86,220,86,0,0,0,0,229,31,179,21,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224] >;

Dic5.4Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5._4{\rm Dic}_6
% in TeX

G:=Group("Dic5.4Dic6");
// GroupNames label

G:=SmallGroup(480,236);
// by ID

G=gap.SmallGroup(480,236);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=a^5*b*c^6,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*b*c^-1>;
// generators/relations

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