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G = C3×C5⋊D4order 120 = 23·3·5

Direct product of C3 and C5⋊D4

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

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

C1C10 — C3×C5⋊D4
C1C5C10C30C6×D5 — C3×C5⋊D4
C5C10 — C3×C5⋊D4
C1C6C2×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 >

2C2
10C2
5C22
5C4
2C6
10C6
2D5
2C10
5D4
5C2×C6
5C12
2C30
2C3×D5
5C3×D4

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 2A2B2C3A3B 4 5A5B6A6B6C6D6E6F10A···10F12A12B15A15B15C15D30A···30L
order12223345566666610···1012121515151530···30
size11210111022112210102···2101022222···2

39 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C3C6C6C6D4D5D10C3×D4C3×D5C5⋊D4C6×D5C3×C5⋊D4
kernelC3×C5⋊D4C3×Dic5C6×D5C2×C30C5⋊D4Dic5D10C2×C10C15C2×C6C6C5C22C3C2C1
# reps1111222212224448

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

250
025
,
2612
2917
,
108
3021
,
120
030
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

Export

Subgroup lattice of C3×C5⋊D4 in TeX

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