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## G = Dic3.F5order 240 = 24·3·5

### The non-split extension by Dic3 of F5 acting via F5/D5=C2

Aliases: Dic3.F5, D30.2C4, C152M4(2), Dic5.10D6, C5⋊C82S3, C15⋊C84C2, C51(C8⋊S3), C31(C4.F5), C2.7(S3×F5), C6.7(C2×F5), C10.7(C4×S3), C30.7(C2×C4), D30.C2.3C2, (C5×Dic3).2C4, (C3×Dic5).10C22, (C3×C5⋊C8)⋊4C2, SmallGroup(240,101)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3.F5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — Dic3.F5
 Lower central C15 — C30 — Dic3.F5
 Upper central C1 — C2

Generators and relations for Dic3.F5
G = < a,b,c,d | a6=c5=1, b2=d4=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c3 >

Character table of Dic3.F5

 class 1 2A 2B 3 4A 4B 4C 5 6 8A 8B 8C 8D 10 12A 12B 15 20A 20B 24A 24B 24C 24D 30 size 1 1 30 2 5 5 6 4 2 10 10 30 30 4 10 10 8 12 12 10 10 10 10 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 1 1 i -i -i i 1 -1 -1 1 -1 -1 -i i i -i 1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 1 i -i i -i 1 -1 -1 1 1 1 -i i i -i 1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 1 -i i -i i 1 -1 -1 1 1 1 i -i -i i 1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 1 -i i i -i 1 -1 -1 1 -1 -1 i -i -i i 1 linear of order 4 ρ9 2 2 0 -1 2 2 0 2 -1 2 2 0 0 2 -1 -1 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 -1 2 2 0 2 -1 -2 -2 0 0 2 -1 -1 -1 0 0 1 1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 0 -1 -2 -2 0 2 -1 2i -2i 0 0 2 1 1 -1 0 0 i -i -i i -1 complex lifted from C4×S3 ρ12 2 -2 0 2 2i -2i 0 2 -2 0 0 0 0 -2 2i -2i 2 0 0 0 0 0 0 -2 complex lifted from M4(2) ρ13 2 -2 0 2 -2i 2i 0 2 -2 0 0 0 0 -2 -2i 2i 2 0 0 0 0 0 0 -2 complex lifted from M4(2) ρ14 2 2 0 -1 -2 -2 0 2 -1 -2i 2i 0 0 2 1 1 -1 0 0 -i i i -i -1 complex lifted from C4×S3 ρ15 2 -2 0 -1 -2i 2i 0 2 1 0 0 0 0 -2 i -i -1 0 0 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 1 complex lifted from C8⋊S3 ρ16 2 -2 0 -1 -2i 2i 0 2 1 0 0 0 0 -2 i -i -1 0 0 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 1 complex lifted from C8⋊S3 ρ17 2 -2 0 -1 2i -2i 0 2 1 0 0 0 0 -2 -i i -1 0 0 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 1 complex lifted from C8⋊S3 ρ18 2 -2 0 -1 2i -2i 0 2 1 0 0 0 0 -2 -i i -1 0 0 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 1 complex lifted from C8⋊S3 ρ19 4 4 0 4 0 0 4 -1 4 0 0 0 0 -1 0 0 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from F5 ρ20 4 4 0 4 0 0 -4 -1 4 0 0 0 0 -1 0 0 -1 1 1 0 0 0 0 -1 orthogonal lifted from C2×F5 ρ21 4 -4 0 4 0 0 0 -1 -4 0 0 0 0 1 0 0 -1 √-5 -√-5 0 0 0 0 1 complex lifted from C4.F5 ρ22 4 -4 0 4 0 0 0 -1 -4 0 0 0 0 1 0 0 -1 -√-5 √-5 0 0 0 0 1 complex lifted from C4.F5 ρ23 8 8 0 -4 0 0 0 -2 -4 0 0 0 0 -2 0 0 1 0 0 0 0 0 0 1 orthogonal lifted from S3×F5 ρ24 8 -8 0 -4 0 0 0 -2 4 0 0 0 0 2 0 0 1 0 0 0 0 0 0 -1 orthogonal faithful

Smallest permutation representation of Dic3.F5
On 120 points
Generators in S120
(1 54 94 5 50 90)(2 55 95 6 51 91)(3 56 96 7 52 92)(4 49 89 8 53 93)(9 37 64 13 33 60)(10 38 57 14 34 61)(11 39 58 15 35 62)(12 40 59 16 36 63)(17 116 48 21 120 44)(18 117 41 22 113 45)(19 118 42 23 114 46)(20 119 43 24 115 47)(25 106 81 29 110 85)(26 107 82 30 111 86)(27 108 83 31 112 87)(28 109 84 32 105 88)(65 102 76 69 98 80)(66 103 77 70 99 73)(67 104 78 71 100 74)(68 97 79 72 101 75)
(1 7 5 3)(2 4 6 8)(9 105 13 109)(10 110 14 106)(11 107 15 111)(12 112 16 108)(17 74 21 78)(18 79 22 75)(19 76 23 80)(20 73 24 77)(25 38 29 34)(26 35 30 39)(27 40 31 36)(28 37 32 33)(41 68 45 72)(42 65 46 69)(43 70 47 66)(44 67 48 71)(49 95 53 91)(50 92 54 96)(51 89 55 93)(52 94 56 90)(57 81 61 85)(58 86 62 82)(59 83 63 87)(60 88 64 84)(97 113 101 117)(98 118 102 114)(99 115 103 119)(100 120 104 116)
(1 39 103 113 28)(2 114 40 29 104)(3 30 115 97 33)(4 98 31 34 116)(5 35 99 117 32)(6 118 36 25 100)(7 26 119 101 37)(8 102 27 38 120)(9 96 86 20 72)(10 21 89 65 87)(11 66 22 88 90)(12 81 67 91 23)(13 92 82 24 68)(14 17 93 69 83)(15 70 18 84 94)(16 85 71 95 19)(41 105 50 62 73)(42 63 106 74 51)(43 75 64 52 107)(44 53 76 108 57)(45 109 54 58 77)(46 59 110 78 55)(47 79 60 56 111)(48 49 80 112 61)
(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)

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

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

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

Dic3.F5 is a maximal subgroup of   S3×C4.F5  D60.C4  Dic6.F5  C5⋊C8⋊D6  C5⋊C8.D6  D15⋊C8⋊C2  D152M4(2)
Dic3.F5 is a maximal quotient of   C30.M4(2)  D30⋊C8  C30.4M4(2)

Matrix representation of Dic3.F5 in GL6(𝔽241)

 240 1 0 0 0 0 240 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 1 0 0 0 0 1 0 0 0 0 0 0 0 17 0 34 34 0 0 207 224 207 0 0 0 0 207 224 207 0 0 34 34 0 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 200 159 208 192 0 0 49 33 82 41 0 0 49 8 208 16 0 0 200 8 233 41

G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,207,0,34,0,0,0,224,207,34,0,0,34,207,224,0,0,0,34,0,207,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,200,49,49,200,0,0,159,33,8,8,0,0,208,82,208,233,0,0,192,41,16,41] >;

Dic3.F5 in GAP, Magma, Sage, TeX

{\rm Dic}_3.F_5
% in TeX

G:=Group("Dic3.F5");
// GroupNames label

G:=SmallGroup(240,101);
// by ID

G=gap.SmallGroup(240,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,50,490,3461,1745]);
// Polycyclic

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

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