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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, C3, C4, C4, C22, C5, C6, C6, C2×C4, Q8, D5, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, Dic6, Dic6, C2×Dic3, C2×C12, C3×D5, C30, C4×Q8, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×F5, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C3⋊F5, C3⋊F5, C6×D5, C4×F5, C4⋊F5, C4⋊F5, Q8×D5, Dic6⋊C4, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, C6×F5, C2×C3⋊F5, Q8×F5, Dic3×F5, Dic3⋊F5, C3×C4⋊F5, C4×C3⋊F5, D5×Dic6, Dic65F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, F5, C4×S3, C22×S3, C4×Q8, C2×F5, 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 15 7 21)(2 14 8 20)(3 13 9 19)(4 24 10 18)(5 23 11 17)(6 22 12 16)(25 44 31 38)(26 43 32 37)(27 42 33 48)(28 41 34 47)(29 40 35 46)(30 39 36 45)(49 71 55 65)(50 70 56 64)(51 69 57 63)(52 68 58 62)(53 67 59 61)(54 66 60 72)(73 96 79 90)(74 95 80 89)(75 94 81 88)(76 93 82 87)(77 92 83 86)(78 91 84 85)(97 119 103 113)(98 118 104 112)(99 117 105 111)(100 116 106 110)(101 115 107 109)(102 114 108 120)
(1 54 36 102 85)(2 55 25 103 86)(3 56 26 104 87)(4 57 27 105 88)(5 58 28 106 89)(6 59 29 107 90)(7 60 30 108 91)(8 49 31 97 92)(9 50 32 98 93)(10 51 33 99 94)(11 52 34 100 95)(12 53 35 101 96)(13 64 43 112 76)(14 65 44 113 77)(15 66 45 114 78)(16 67 46 115 79)(17 68 47 116 80)(18 69 48 117 81)(19 70 37 118 82)(20 71 38 119 83)(21 72 39 120 84)(22 61 40 109 73)(23 62 41 110 74)(24 63 42 111 75)
(1 21 7 15)(2 16 8 22)(3 23 9 17)(4 18 10 24)(5 13 11 19)(6 20 12 14)(25 79 97 61)(26 74 98 68)(27 81 99 63)(28 76 100 70)(29 83 101 65)(30 78 102 72)(31 73 103 67)(32 80 104 62)(33 75 105 69)(34 82 106 64)(35 77 107 71)(36 84 108 66)(37 89 112 52)(38 96 113 59)(39 91 114 54)(40 86 115 49)(41 93 116 56)(42 88 117 51)(43 95 118 58)(44 90 119 53)(45 85 120 60)(46 92 109 55)(47 87 110 50)(48 94 111 57)

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

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

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

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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