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G = D5xDic6order 240 = 24·3·5

Direct product of D5 and Dic6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5xDic6, C20.13D6, Dic30:9C2, D10.16D6, C12.26D10, C30.1C23, C60.19C22, Dic5.11D6, Dic3.1D10, Dic15.1C22, (C3xD5):Q8, C3:2(Q8xD5), C15:Q8:1C2, C15:1(C2xQ8), C5:1(C2xDic6), (C4xD5).1S3, C4.12(S3xD5), (C5xDic6):2C2, (D5xC12).1C2, C6.1(C22xD5), C10.1(C22xS3), (D5xDic3).1C2, (C6xD5).9C22, (C5xDic3).1C22, (C3xDic5).11C22, C2.5(C2xS3xD5), SmallGroup(240,125)

Series: Derived Chief Lower central Upper central

C1C30 — D5xDic6
C1C5C15C30C6xD5D5xDic3 — D5xDic6
C15C30 — D5xDic6
C1C2C4

Generators and relations for D5xDic6
 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, C2xC4, Q8, D5, C10, Dic3, Dic3, C12, C12, C2xC6, C15, C2xQ8, Dic5, Dic5, C20, C20, D10, Dic6, Dic6, C2xDic3, C2xC12, C3xD5, C30, Dic10, C4xD5, C4xD5, C5xQ8, C2xDic6, C5xDic3, C3xDic5, Dic15, C60, C6xD5, Q8xD5, D5xDic3, C15:Q8, D5xC12, C5xDic6, Dic30, D5xDic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2xQ8, D10, Dic6, C22xS3, C22xD5, C2xDic6, S3xD5, Q8xD5, C2xS3xD5, D5xDic6

Smallest permutation representation of D5xDic6
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)]])

D5xDic6 is a maximal subgroup of
Dic6:F5  Dic30:C4  Dic60:C2  C24.2D10  C60.8C23  D20.13D6  Dic6:5F5  D20.38D6  D20.39D6  C15:2- 1+4  D20.29D6  S3xQ8xD5
D5xDic6 is a maximal quotient of
Dic5:5Dic6  Dic3:Dic10  Dic15:Q8  Dic30:17C4  Dic5.1Dic6  Dic5.2Dic6  D10:Dic6  C60.67D4  C60.68D4  Dic5:Dic6  Dic5.7Dic6  D10:1Dic6  D10:2Dic6  Dic15.D4  D10:4Dic6  C60:Q8

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C10A10B12A12B12C12D15A15B20A20B20C20D20E20F30A30B60A60B60C60D
order12223444444556661010121212121515202020202020303060606060
size11552266103030222101022221010444412121212444444

36 irreducible representations

dim1111112222222224444
type+++++++-++++++-+-+-
imageC1C2C2C2C2C2S3Q8D5D6D6D6D10D10Dic6S3xD5Q8xD5C2xS3xD5D5xDic6
kernelD5xDic6D5xDic3C15:Q8D5xC12C5xDic6Dic30C4xD5C3xD5Dic6Dic5C20D10Dic3C12D5C4C3C2C1
# reps1221111221114242224

Matrix representation of D5xDic6 in GL4(F61) generated by

43100
60000
0010
0001
,
14300
06000
00600
00060
,
60000
06000
005340
00320
,
1000
0100
005823
00423
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] >;

D5xDic6 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

׿
x
:
Z
F
o
wr
Q
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