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## G = D5×C12order 120 = 23·3·5

### Direct product of C12 and D5

Aliases: D5×C12, C202C6, C605C2, C12Dic5, Dic52C6, D10.2C6, C6.14D10, C30.14C22, C52(C2×C12), C158(C2×C4), C4(C3×Dic5), C2.1(C6×D5), C10.2(C2×C6), C12(C3×Dic5), (C6×D5).4C2, (C3×Dic5)⋊5C2, SmallGroup(120,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C12
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C12
 Lower central C5 — D5×C12
 Upper central C1 — C12

Generators and relations for D5×C12
G = < a,b,c | a12=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C12
On 60 points
Generators in S60
(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)
(1 55 27 16 39)(2 56 28 17 40)(3 57 29 18 41)(4 58 30 19 42)(5 59 31 20 43)(6 60 32 21 44)(7 49 33 22 45)(8 50 34 23 46)(9 51 35 24 47)(10 52 36 13 48)(11 53 25 14 37)(12 54 26 15 38)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 37)(12 38)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 49)(23 50)(24 51)

G:=sub<Sym(60)| (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), (1,55,27,16,39)(2,56,28,17,40)(3,57,29,18,41)(4,58,30,19,42)(5,59,31,20,43)(6,60,32,21,44)(7,49,33,22,45)(8,50,34,23,46)(9,51,35,24,47)(10,52,36,13,48)(11,53,25,14,37)(12,54,26,15,38), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51)>;

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), (1,55,27,16,39)(2,56,28,17,40)(3,57,29,18,41)(4,58,30,19,42)(5,59,31,20,43)(6,60,32,21,44)(7,49,33,22,45)(8,50,34,23,46)(9,51,35,24,47)(10,52,36,13,48)(11,53,25,14,37)(12,54,26,15,38), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,49)(23,50)(24,51) );

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)], [(1,55,27,16,39),(2,56,28,17,40),(3,57,29,18,41),(4,58,30,19,42),(5,59,31,20,43),(6,60,32,21,44),(7,49,33,22,45),(8,50,34,23,46),(9,51,35,24,47),(10,52,36,13,48),(11,53,25,14,37),(12,54,26,15,38)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,37),(12,38),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,49),(23,50),(24,51)]])

D5×C12 is a maximal subgroup of   C20.32D6  C60.C4  C12.F5  C60⋊C4  D6.D10  D125D5  C12.28D10

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D5 D10 C3×D5 C4×D5 C6×D5 D5×C12 kernel D5×C12 C3×Dic5 C60 C6×D5 C4×D5 C3×D5 Dic5 C20 D10 D5 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 2 2 4 4 4 8

Matrix representation of D5×C12 in GL2(𝔽61) generated by

 21 0 0 21
,
 0 1 60 17
,
 1 0 17 60
G:=sub<GL(2,GF(61))| [21,0,0,21],[0,60,1,17],[1,17,0,60] >;

D5×C12 in GAP, Magma, Sage, TeX

D_5\times C_{12}
% in TeX

G:=Group("D5xC12");
// GroupNames label

G:=SmallGroup(120,17);
// by ID

G=gap.SmallGroup(120,17);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-5,66,2404]);
// Polycyclic

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

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