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G = Dic6⋊F5order 480 = 25·3·5

3rd semidirect product of Dic6 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic63F5, Dic301C4, Dic5.2D12, C4⋊F5.1S3, C4.9(S3×F5), C20.3(C4×S3), C60.2(C2×C4), C5⋊(C6.SD16), C12.2(C2×F5), (C5×Dic6)⋊1C4, (C6×D5).19D4, C32(Q8⋊F5), (C4×D5).21D6, C2.6(D6⋊F5), (C3×D5).1Q16, C10.3(D6⋊C4), C151(Q8⋊C4), (D5×Dic6).3C2, (C3×D5).3SD16, C6.3(C22⋊F5), C60.C4.1C2, C30.3(C22⋊C4), D5.2(D4.S3), (C3×Dic5).22D4, D5.2(C3⋊Q16), D10.23(C3⋊D4), (D5×C12).31C22, (C3×C4⋊F5).1C2, SmallGroup(480,229)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic6⋊F5
 G = < a,b,c,d | a12=c5=d4=1, b2=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c3 >

Subgroups: 468 in 84 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22, C5, C6, C6 [×2], C8, C2×C4 [×3], Q8 [×3], D5 [×2], C10, Dic3 [×2], C12, C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C3⋊C8, Dic6, Dic6 [×2], C2×Dic3, C2×C12 [×2], C3×D5 [×2], C30, Q8⋊C4, C5⋊C8, Dic10 [×2], C4×D5, C4×D5, C5×Q8, C2×F5, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, C6.SD16, C15⋊C8, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, C6×F5, Q8⋊F5, C3×C4⋊F5, C60.C4, D5×Dic6, Dic6⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, SD16, Q16, F5, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×F5, D6⋊C4, D4.S3, C3⋊Q16, C22⋊F5, C6.SD16, S3×F5, Q8⋊F5, D6⋊F5, Dic6⋊F5

Smallest permutation representation of Dic6⋊F5
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 40 7 46)(2 39 8 45)(3 38 9 44)(4 37 10 43)(5 48 11 42)(6 47 12 41)(13 93 19 87)(14 92 20 86)(15 91 21 85)(16 90 22 96)(17 89 23 95)(18 88 24 94)(25 55 31 49)(26 54 32 60)(27 53 33 59)(28 52 34 58)(29 51 35 57)(30 50 36 56)(61 117 67 111)(62 116 68 110)(63 115 69 109)(64 114 70 120)(65 113 71 119)(66 112 72 118)(73 107 79 101)(74 106 80 100)(75 105 81 99)(76 104 82 98)(77 103 83 97)(78 102 84 108)
(1 85 57 109 74)(2 86 58 110 75)(3 87 59 111 76)(4 88 60 112 77)(5 89 49 113 78)(6 90 50 114 79)(7 91 51 115 80)(8 92 52 116 81)(9 93 53 117 82)(10 94 54 118 83)(11 95 55 119 84)(12 96 56 120 73)(13 27 61 104 38)(14 28 62 105 39)(15 29 63 106 40)(16 30 64 107 41)(17 31 65 108 42)(18 32 66 97 43)(19 33 67 98 44)(20 34 68 99 45)(21 35 69 100 46)(22 36 70 101 47)(23 25 71 102 48)(24 26 72 103 37)
(2 8)(4 10)(6 12)(13 36 104 70)(14 31 105 65)(15 26 106 72)(16 33 107 67)(17 28 108 62)(18 35 97 69)(19 30 98 64)(20 25 99 71)(21 32 100 66)(22 27 101 61)(23 34 102 68)(24 29 103 63)(37 40)(38 47)(39 42)(41 44)(43 46)(45 48)(49 78 113 89)(50 73 114 96)(51 80 115 91)(52 75 116 86)(53 82 117 93)(54 77 118 88)(55 84 119 95)(56 79 120 90)(57 74 109 85)(58 81 110 92)(59 76 111 87)(60 83 112 94)

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

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

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

33 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D 10 12A12B···12F 15 20A20B20C 30 60A60B
order1222344444456668888101212···1215202020306060
size1155221012202060421010303030304420···20882424888

33 irreducible representations

dim111111222222222444448888
type++++++++-+++--++-+-
imageC1C2C2C2C4C4S3D4D4D6SD16Q16D12C4×S3C3⋊D4F5C2×F5D4.S3C3⋊Q16C22⋊F5S3×F5Q8⋊F5D6⋊F5Dic6⋊F5
kernelDic6⋊F5C3×C4⋊F5C60.C4D5×Dic6C5×Dic6Dic30C4⋊F5C3×Dic5C6×D5C4×D5C3×D5C3×D5Dic5C20D10Dic6C12D5D5C6C4C3C2C1
# reps111122111122222111121112

Matrix representation of Dic6⋊F5 in GL8(𝔽241)

2405000000
961000000
001500000
00102250000
0000240000
0000024000
0000002400
0000000240
,
0146000000
1370000000
0059340000
001671820000
00001170234234
0000712470
0000071247
00002342340117
,
10000000
01000000
00100000
00010000
0000240240240240
00001000
00000100
00000010
,
640000000
122177000000
006400000
000640000
00001000
00000001
00000100
0000240240240240

G:=sub<GL(8,GF(241))| [240,96,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,15,10,0,0,0,0,0,0,0,225,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[0,137,0,0,0,0,0,0,146,0,0,0,0,0,0,0,0,0,59,167,0,0,0,0,0,0,34,182,0,0,0,0,0,0,0,0,117,7,0,234,0,0,0,0,0,124,7,234,0,0,0,0,234,7,124,0,0,0,0,0,234,0,7,117],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,1,0,0,0,0,240,0,0,0],[64,122,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240] >;

Dic6⋊F5 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes F_5
% in TeX

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

G:=SmallGroup(480,229);
// by ID

G=gap.SmallGroup(480,229);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,1356,9414,4724]);
// 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,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;
// generators/relations

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