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## G = D5×Dic6order 240 = 24·3·5

### Direct product of D5 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×Dic6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — D5×Dic6
 Lower central C15 — C30 — D5×Dic6
 Upper central C1 — C2 — C4

Generators and relations for D5×Dic6
G = < a,b,c,d | a5=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 288 in 76 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, Q8, D5, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C2×Q8, Dic5, Dic5, C20, C20, D10, Dic6, Dic6, C2×Dic3, C2×C12, C3×D5, C30, Dic10, C4×D5, C4×D5, C5×Q8, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, Q8×D5, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, D5×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, D10, Dic6, C22×S3, C22×D5, C2×Dic6, S3×D5, Q8×D5, C2×S3×D5, D5×Dic6

Smallest permutation representation of D5×Dic6
On 120 points
Generators in S120
(1 105 116 81 63)(2 106 117 82 64)(3 107 118 83 65)(4 108 119 84 66)(5 97 120 73 67)(6 98 109 74 68)(7 99 110 75 69)(8 100 111 76 70)(9 101 112 77 71)(10 102 113 78 72)(11 103 114 79 61)(12 104 115 80 62)(13 94 51 34 41)(14 95 52 35 42)(15 96 53 36 43)(16 85 54 25 44)(17 86 55 26 45)(18 87 56 27 46)(19 88 57 28 47)(20 89 58 29 48)(21 90 59 30 37)(22 91 60 31 38)(23 92 49 32 39)(24 93 50 33 40)
(1 69)(2 70)(3 71)(4 72)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 85)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(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 59 7 53)(2 58 8 52)(3 57 9 51)(4 56 10 50)(5 55 11 49)(6 54 12 60)(13 83 19 77)(14 82 20 76)(15 81 21 75)(16 80 22 74)(17 79 23 73)(18 78 24 84)(25 104 31 98)(26 103 32 97)(27 102 33 108)(28 101 34 107)(29 100 35 106)(30 99 36 105)(37 110 43 116)(38 109 44 115)(39 120 45 114)(40 119 46 113)(41 118 47 112)(42 117 48 111)(61 92 67 86)(62 91 68 85)(63 90 69 96)(64 89 70 95)(65 88 71 94)(66 87 72 93)

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

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

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

D5×Dic6 is a maximal subgroup of
Dic6⋊F5  Dic30⋊C4  Dic60⋊C2  C24.2D10  C60.8C23  D20.13D6  Dic65F5  D20.38D6  D20.39D6  C15⋊2- 1+4  D20.29D6  S3×Q8×D5
D5×Dic6 is a maximal quotient of
Dic55Dic6  Dic3⋊Dic10  Dic15⋊Q8  Dic3017C4  Dic5.1Dic6  Dic5.2Dic6  D10⋊Dic6  C60.67D4  C60.68D4  Dic5⋊Dic6  Dic5.7Dic6  D101Dic6  D102Dic6  Dic15.D4  D104Dic6  C60⋊Q8

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D6 D10 D10 Dic6 S3×D5 Q8×D5 C2×S3×D5 D5×Dic6 kernel D5×Dic6 D5×Dic3 C15⋊Q8 D5×C12 C5×Dic6 Dic30 C4×D5 C3×D5 Dic6 Dic5 C20 D10 Dic3 C12 D5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 2 1 1 1 4 2 4 2 2 2 4

Matrix representation of D5×Dic6 in GL4(𝔽61) generated by

 43 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 1 43 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 53 40 0 0 32 0
,
 1 0 0 0 0 1 0 0 0 0 58 23 0 0 42 3
G:=sub<GL(4,GF(61))| [43,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,43,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,53,32,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,58,42,0,0,23,3] >;

D5×Dic6 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_6
% in TeX

G:=Group("D5xDic6");
// GroupNames label

G:=SmallGroup(240,125);
// by ID

G=gap.SmallGroup(240,125);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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