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G = Dic65F5order 480 = 25·3·5

5th semidirect product of Dic6 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic65F5, Dic303C4, C3⋊F5⋊Q8, C32(Q8×F5), C15⋊Q82C4, C152(C4×Q8), C4⋊F5.2S3, C60.5(C2×C4), (C2×F5).2D6, C4.12(S3×F5), D5.1(S3×Q8), C20.10(C4×S3), C5⋊(Dic6⋊C4), (C5×Dic6)⋊3C4, (C4×D5).32D6, C12.14(C2×F5), C6.3(C22×F5), Dic3⋊F5.2C2, C30.3(C22×C4), (D5×Dic6).4C2, Dic5.2(C4×S3), Dic3.2(C2×F5), (Dic3×F5).2C2, (C6×F5).2C22, Dic15.2(C2×C4), (C6×D5).23C23, D5.1(D42S3), D10.26(C22×S3), (D5×C12).34C22, (D5×Dic3).6C22, C2.8(C2×S3×F5), C10.3(S3×C2×C4), (C4×C3⋊F5).1C2, (C3×C4⋊F5).2C2, (C3×D5).2(C2×Q8), (C2×C3⋊F5).7C22, (C3×D5).3(C4○D4), (C5×Dic3).2(C2×C4), (C3×Dic5).21(C2×C4), SmallGroup(480,984)

Series: Derived Chief Lower central Upper central

C1C30 — Dic65F5
C1C5C15C3×D5C6×D5C6×F5Dic3×F5 — Dic65F5
C15C30 — Dic65F5
C1C2C4

Generators and relations for Dic65F5
 G = < a,b,c,d | a12=c5=d4=1, b2=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 660 in 140 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×10], C22, C5, C6, C6 [×2], C2×C4 [×7], Q8 [×4], D5 [×2], C10, Dic3 [×2], Dic3 [×5], C12, C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5, Dic5 [×2], C20, C20 [×2], F5 [×5], D10, Dic6, Dic6 [×3], C2×Dic3 [×4], C2×C12 [×3], C3×D5 [×2], C30, C4×Q8, Dic10 [×3], C4×D5, C4×D5 [×2], C5×Q8, C2×F5 [×2], C2×F5 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60, C3×F5 [×2], C3⋊F5 [×2], C3⋊F5, C6×D5, C4×F5 [×3], C4⋊F5, C4⋊F5 [×2], Q8×D5, Dic6⋊C4, D5×Dic3 [×2], C15⋊Q8 [×2], D5×C12, C5×Dic6, Dic30, C6×F5 [×2], C2×C3⋊F5 [×2], Q8×F5, Dic3×F5 [×2], Dic3⋊F5 [×2], C3×C4⋊F5, C4×C3⋊F5, D5×Dic6, Dic65F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D6 [×3], C22×C4, C2×Q8, C4○D4, F5, C4×S3 [×2], C22×S3, C4×Q8, C2×F5 [×3], S3×C2×C4, D42S3, S3×Q8, C22×F5, Dic6⋊C4, S3×F5, Q8×F5, C2×S3×F5, Dic65F5

Smallest permutation representation of Dic65F5
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 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 52 19 58)(14 51 20 57)(15 50 21 56)(16 49 22 55)(17 60 23 54)(18 59 24 53)(25 112 31 118)(26 111 32 117)(27 110 33 116)(28 109 34 115)(29 120 35 114)(30 119 36 113)(61 107 67 101)(62 106 68 100)(63 105 69 99)(64 104 70 98)(65 103 71 97)(66 102 72 108)(73 90 79 96)(74 89 80 95)(75 88 81 94)(76 87 82 93)(77 86 83 92)(78 85 84 91)
(1 99 15 80 109)(2 100 16 81 110)(3 101 17 82 111)(4 102 18 83 112)(5 103 19 84 113)(6 104 20 73 114)(7 105 21 74 115)(8 106 22 75 116)(9 107 23 76 117)(10 108 24 77 118)(11 97 13 78 119)(12 98 14 79 120)(25 45 66 53 86)(26 46 67 54 87)(27 47 68 55 88)(28 48 69 56 89)(29 37 70 57 90)(30 38 71 58 91)(31 39 72 59 92)(32 40 61 60 93)(33 41 62 49 94)(34 42 63 50 95)(35 43 64 51 96)(36 44 65 52 85)
(1 48 7 42)(2 43 8 37)(3 38 9 44)(4 45 10 39)(5 40 11 46)(6 47 12 41)(13 26 84 61)(14 33 73 68)(15 28 74 63)(16 35 75 70)(17 30 76 65)(18 25 77 72)(19 32 78 67)(20 27 79 62)(21 34 80 69)(22 29 81 64)(23 36 82 71)(24 31 83 66)(49 114 88 98)(50 109 89 105)(51 116 90 100)(52 111 91 107)(53 118 92 102)(54 113 93 97)(55 120 94 104)(56 115 95 99)(57 110 96 106)(58 117 85 101)(59 112 86 108)(60 119 87 103)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D···4H4I4J4K4L4M4N4O4P 5 6A6B6C 10 12A12B···12F 15 20A20B20C 30 60A60B
order122234444···4444444445666101212···1215202020306060
size1155226610···1015151515303030304210104420···20882424888

39 irreducible representations

dim1111111112222222444448888
type+++++++-+++++--+-+-
imageC1C2C2C2C2C2C4C4C4S3Q8D6D6C4○D4C4×S3C4×S3F5C2×F5C2×F5D42S3S3×Q8S3×F5Q8×F5C2×S3×F5Dic65F5
kernelDic65F5Dic3×F5Dic3⋊F5C3×C4⋊F5C4×C3⋊F5D5×Dic6C15⋊Q8C5×Dic6Dic30C4⋊F5C3⋊F5C4×D5C2×F5C3×D5Dic5C20Dic6Dic3C12D5D5C4C3C2C1
# reps1221114221212222121111112

Matrix representation of Dic65F5 in GL8(𝔽61)

601000000
600000000
000600000
00100000
000060000
000006000
000000600
000000060
,
01000000
10000000
0043380000
0038180000
000060000
000006000
000000600
000000060
,
10000000
01000000
00100000
00010000
000060606060
00001000
00000100
00000010
,
500000000
050000000
001590000
009460000
000060000
000000060
000006000
00001111

G:=sub<GL(8,GF(61))| [60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,43,38,0,0,0,0,0,0,38,18,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0],[50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,15,9,0,0,0,0,0,0,9,46,0,0,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,60,0,1] >;

Dic65F5 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_5F_5
% in TeX

G:=Group("Dic6:5F5");
// GroupNames label

G:=SmallGroup(480,984);
// by ID

G=gap.SmallGroup(480,984);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,219,100,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^5=d^4=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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