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## G = C3×C5⋊D4order 120 = 23·3·5

### Direct product of C3 and C5⋊D4

Aliases: C3×C5⋊D4, C158D4, Dic5⋊C6, D102C6, C6.17D10, C30.17C22, C52(C3×D4), (C2×C6)⋊1D5, (C2×C30)⋊4C2, (C2×C10)⋊4C6, (C6×D5)⋊5C2, C2.5(C6×D5), C10.5(C2×C6), C222(C3×D5), (C3×Dic5)⋊4C2, SmallGroup(120,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C5⋊D4
 Chief series C1 — C5 — C10 — C30 — C6×D5 — C3×C5⋊D4
 Lower central C5 — C10 — C3×C5⋊D4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C5⋊D4
G = < a,b,c,d | a3=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C3×C5⋊D4
On 60 points
Generators in S60
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(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 31 6 36)(2 35 7 40)(3 34 8 39)(4 33 9 38)(5 32 10 37)(11 41 16 46)(12 45 17 50)(13 44 18 49)(14 43 19 48)(15 42 20 47)(21 51 26 56)(22 55 27 60)(23 54 28 59)(24 53 29 58)(25 52 30 57)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)

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

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

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

C3×C5⋊D4 is a maximal subgroup of   Dic5.D6  C30.C23  D10⋊D6  C3×D4×D5  SL2(𝔽3).11D10

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D10 C3×D4 C3×D5 C5⋊D4 C6×D5 C3×C5⋊D4 kernel C3×C5⋊D4 C3×Dic5 C6×D5 C2×C30 C5⋊D4 Dic5 D10 C2×C10 C15 C2×C6 C6 C5 C22 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 4 4 4 8

Matrix representation of C3×C5⋊D4 in GL2(𝔽31) generated by

 25 0 0 25
,
 26 12 29 17
,
 10 8 30 21
,
 1 20 0 30
G:=sub<GL(2,GF(31))| [25,0,0,25],[26,29,12,17],[10,30,8,21],[1,0,20,30] >;

C3×C5⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes D_4
% in TeX

G:=Group("C3xC5:D4");
// GroupNames label

G:=SmallGroup(120,20);
// by ID

G=gap.SmallGroup(120,20);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-5,141,2404]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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