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## G = Dic6⋊F5order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic6⋊F5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C3×C4⋊F5 — Dic6⋊F5
 Lower central C15 — C30 — C60 — Dic6⋊F5
 Upper central C1 — C2 — C4

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 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 8A 8B 8C 8D 10 12A 12B ··· 12F 15 20A 20B 20C 30 60A 60B order 1 2 2 2 3 4 4 4 4 4 4 5 6 6 6 8 8 8 8 10 12 12 ··· 12 15 20 20 20 30 60 60 size 1 1 5 5 2 2 10 12 20 20 60 4 2 10 10 30 30 30 30 4 4 20 ··· 20 8 8 24 24 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 8 8 8 8 type + + + + + + + + - + + + - - + + - + - image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 SD16 Q16 D12 C4×S3 C3⋊D4 F5 C2×F5 D4.S3 C3⋊Q16 C22⋊F5 S3×F5 Q8⋊F5 D6⋊F5 Dic6⋊F5 kernel Dic6⋊F5 C3×C4⋊F5 C60.C4 D5×Dic6 C5×Dic6 Dic30 C4⋊F5 C3×Dic5 C6×D5 C4×D5 C3×D5 C3×D5 Dic5 C20 D10 Dic6 C12 D5 D5 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 1 1 1 1 2 1 1 1 2

Matrix representation of Dic6⋊F5 in GL8(𝔽241)

 240 5 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 10 225 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
,
 0 146 0 0 0 0 0 0 137 0 0 0 0 0 0 0 0 0 59 34 0 0 0 0 0 0 167 182 0 0 0 0 0 0 0 0 117 0 234 234 0 0 0 0 7 124 7 0 0 0 0 0 0 7 124 7 0 0 0 0 234 234 0 117
,
 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 0 240 240 240 240 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 64 0 0 0 0 0 0 0 122 177 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240

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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