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## G = D4×C15order 120 = 23·3·5

### Direct product of C15 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C15, C4⋊C30, C607C2, C203C6, C123C10, C222C30, C30.23C22, (C2×C6)⋊1C10, (C2×C30)⋊1C2, (C2×C10)⋊3C6, C10.6(C2×C6), C6.6(C2×C10), C2.1(C2×C30), SmallGroup(120,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C15
 Chief series C1 — C2 — C10 — C30 — C2×C30 — D4×C15
 Lower central C1 — C2 — D4×C15
 Upper central C1 — C30 — D4×C15

Generators and relations for D4×C15
G = < a,b,c | a15=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D4×C15
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 25 37 55)(2 26 38 56)(3 27 39 57)(4 28 40 58)(5 29 41 59)(6 30 42 60)(7 16 43 46)(8 17 44 47)(9 18 45 48)(10 19 31 49)(11 20 32 50)(12 21 33 51)(13 22 34 52)(14 23 35 53)(15 24 36 54)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 43)(17 44)(18 45)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)

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

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

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

D4×C15 is a maximal subgroup of   D4⋊D15  D4.D15  D42D15

75 conjugacy classes

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

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 D4 C3×D4 C5×D4 D4×C15 kernel D4×C15 C60 C2×C30 C5×D4 C3×D4 C20 C2×C10 C12 C2×C6 D4 C4 C22 C15 C5 C3 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 2 4 8

Matrix representation of D4×C15 in GL2(𝔽31) generated by

 19 0 0 19
,
 0 24 9 0
,
 1 0 0 30
G:=sub<GL(2,GF(31))| [19,0,0,19],[0,9,24,0],[1,0,0,30] >;

D4×C15 in GAP, Magma, Sage, TeX

D_4\times C_{15}
% in TeX

G:=Group("D4xC15");
// GroupNames label

G:=SmallGroup(120,32);
// by ID

G=gap.SmallGroup(120,32);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-2,621]);
// Polycyclic

G:=Group<a,b,c|a^15=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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