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G = Dic3.F5order 240 = 24·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C30 — Dic3.F5
C1C5C15C30C3×Dic5C3×C5⋊C8 — Dic3.F5
C15C30 — Dic3.F5
C1C2

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 >

30C2
3C4
5C4
15C22
10S3
6D5
5C8
15C8
15C2×C4
5D6
5C12
3D10
3C20
2D15
15M4(2)
5C24
5C3⋊C8
5C4×S3
3C5⋊C8
3C4×D5
5C8⋊S3
3C4.F5

Character table of Dic3.F5

 class 12A2B34A4B4C568A8B8C8D1012A12B1520A20B24A24B24C24D30
 size 1130255642101030304101081212101010108
ρ1111111111111111111111111    trivial
ρ211-1111-11111-1-11111-1-111111    linear of order 2
ρ311-1111-111-1-1111111-1-1-1-1-1-11    linear of order 2
ρ4111111111-1-1-1-1111111-1-1-1-11    linear of order 2
ρ51111-1-1-111i-i-ii1-1-11-1-1-iii-i1    linear of order 4
ρ611-11-1-1111i-ii-i1-1-1111-iii-i1    linear of order 4
ρ711-11-1-1111-ii-ii1-1-1111i-i-ii1    linear of order 4
ρ81111-1-1-111-iii-i1-1-11-1-1i-i-ii1    linear of order 4
ρ9220-12202-122002-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ10220-12202-1-2-2002-1-1-1001111-1    orthogonal lifted from D6
ρ11220-1-2-202-12i-2i00211-100i-i-ii-1    complex lifted from C4×S3
ρ122-2022i-2i02-20000-22i-2i2000000-2    complex lifted from M4(2)
ρ132-202-2i2i02-20000-2-2i2i2000000-2    complex lifted from M4(2)
ρ14220-1-2-202-1-2i2i00211-100-iii-i-1    complex lifted from C4×S3
ρ152-20-1-2i2i0210000-2i-i-10085ζ38583ζ38387ζ3878ζ381    complex lifted from C8⋊S3
ρ162-20-1-2i2i0210000-2i-i-1008ζ3887ζ38783ζ38385ζ3851    complex lifted from C8⋊S3
ρ172-20-12i-2i0210000-2-ii-10083ζ38385ζ3858ζ3887ζ3871    complex lifted from C8⋊S3
ρ182-20-12i-2i0210000-2-ii-10087ζ3878ζ3885ζ38583ζ3831    complex lifted from C8⋊S3
ρ194404004-140000-100-1-1-10000-1    orthogonal lifted from F5
ρ20440400-4-140000-100-1110000-1    orthogonal lifted from C2×F5
ρ214-404000-1-40000100-1-5--500001    complex lifted from C4.F5
ρ224-404000-1-40000100-1--5-500001    complex lifted from C4.F5
ρ23880-4000-2-40000-20010000001    orthogonal lifted from S3×F5
ρ248-80-4000-2400002001000000-1    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)

24010000
24000000
00240000
00024000
00002400
00000240
,
010000
100000
001703434
002072242070
000207224207
003434017
,
100000
010000
00240240240240
001000
000100
000010
,
100000
010000
00200159208192
0049338241
0049820816
00200823341

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

Export

Subgroup lattice of Dic3.F5 in TeX
Character table of Dic3.F5 in TeX

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