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G = C5×Dic6order 120 = 23·3·5

Direct product of C5 and Dic6

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

Aliases: C5×Dic6, C154Q8, C60.4C2, C20.3S3, C12.1C10, C10.13D6, Dic3.C10, C30.18C22, C3⋊(C5×Q8), C4.(C5×S3), C2.3(S3×C10), C6.1(C2×C10), (C5×Dic3).2C2, SmallGroup(120,21)

Series: Derived Chief Lower central Upper central

C1C6 — C5×Dic6
C1C3C6C30C5×Dic3 — C5×Dic6
C3C6 — C5×Dic6
C1C10C20

Generators and relations for C5×Dic6
 G = < a,b,c | a5=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3Q8
3C20
3C20
3C5×Q8

Smallest permutation representation of C5×Dic6
Regular action on 120 points
Generators in S120
(1 119 95 55 33)(2 120 96 56 34)(3 109 85 57 35)(4 110 86 58 36)(5 111 87 59 25)(6 112 88 60 26)(7 113 89 49 27)(8 114 90 50 28)(9 115 91 51 29)(10 116 92 52 30)(11 117 93 53 31)(12 118 94 54 32)(13 68 84 42 107)(14 69 73 43 108)(15 70 74 44 97)(16 71 75 45 98)(17 72 76 46 99)(18 61 77 47 100)(19 62 78 48 101)(20 63 79 37 102)(21 64 80 38 103)(22 65 81 39 104)(23 66 82 40 105)(24 67 83 41 106)
(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 20 7 14)(2 19 8 13)(3 18 9 24)(4 17 10 23)(5 16 11 22)(6 15 12 21)(25 98 31 104)(26 97 32 103)(27 108 33 102)(28 107 34 101)(29 106 35 100)(30 105 36 99)(37 49 43 55)(38 60 44 54)(39 59 45 53)(40 58 46 52)(41 57 47 51)(42 56 48 50)(61 115 67 109)(62 114 68 120)(63 113 69 119)(64 112 70 118)(65 111 71 117)(66 110 72 116)(73 95 79 89)(74 94 80 88)(75 93 81 87)(76 92 82 86)(77 91 83 85)(78 90 84 96)

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

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

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

C5×Dic6 is a maximal subgroup of
C30.D4  Dic6⋊D5  C15⋊Q16  C5⋊Dic12  D20⋊S3  D15⋊Q8  C12.28D10  C5×S3×Q8

45 conjugacy classes

class 1  2  3 4A4B4C5A5B5C5D 6 10A10B10C10D12A12B15A15B15C15D20A20B20C20D20E···20L30A30B30C30D60A···60H
order12344455556101010101212151515152020202020···203030303060···60
size11226611112111122222222226···622222···2

45 irreducible representations

dim11111122222222
type++++-+-
imageC1C2C2C5C10C10S3Q8D6Dic6C5×S3C5×Q8S3×C10C5×Dic6
kernelC5×Dic6C5×Dic3C60Dic6Dic3C12C20C15C10C5C4C3C2C1
# reps12148411124448

Matrix representation of C5×Dic6 in GL2(𝔽11) generated by

90
09
,
12
74
,
1010
21
G:=sub<GL(2,GF(11))| [9,0,0,9],[1,7,2,4],[10,2,10,1] >;

C5×Dic6 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_6
% in TeX

G:=Group("C5xDic6");
// GroupNames label

G:=SmallGroup(120,21);
// by ID

G=gap.SmallGroup(120,21);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-3,100,221,106,2004]);
// Polycyclic

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

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Subgroup lattice of C5×Dic6 in TeX

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