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G = D5×C2×C6order 120 = 23·3·5

Direct product of C2×C6 and D5

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

Aliases: D5×C2×C6, C153C23, C303C22, C10⋊(C2×C6), C5⋊(C22×C6), (C2×C10)⋊5C6, (C2×C30)⋊5C2, SmallGroup(120,44)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C2×C6
C1C5C15C3×D5C6×D5 — D5×C2×C6
C5 — D5×C2×C6
C1C2×C6

Generators and relations for D5×C2×C6
 G = < a,b,c,d | a2=b6=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 152 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3, C22, C22 [×6], C5, C6 [×3], C6 [×4], C23, D5 [×4], C10 [×3], C2×C6, C2×C6 [×6], C15, D10 [×6], C2×C10, C22×C6, C3×D5 [×4], C30 [×3], C22×D5, C6×D5 [×6], C2×C30, D5×C2×C6
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, D5, C2×C6 [×7], D10 [×3], C22×C6, C3×D5, C22×D5, C6×D5 [×3], D5×C2×C6

Smallest permutation representation of D5×C2×C6
On 60 points
Generators in S60
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 44)(8 45)(9 46)(10 47)(11 48)(12 43)(13 50)(14 51)(15 52)(16 53)(17 54)(18 49)(19 36)(20 31)(21 32)(22 33)(23 34)(24 35)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)
(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 36 44 53 38)(2 31 45 54 39)(3 32 46 49 40)(4 33 47 50 41)(5 34 48 51 42)(6 35 43 52 37)(7 16 27 59 19)(8 17 28 60 20)(9 18 29 55 21)(10 13 30 56 22)(11 14 25 57 23)(12 15 26 58 24)
(1 41)(2 42)(3 37)(4 38)(5 39)(6 40)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 60)(26 55)(27 56)(28 57)(29 58)(30 59)(31 51)(32 52)(33 53)(34 54)(35 49)(36 50)(43 46)(44 47)(45 48)

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

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

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

D5×C2×C6 is a maximal subgroup of   D10⋊Dic3  D10.D6

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B5A5B6A···6F6G···6N10A···10F15A15B15C15D30A···30L
order1222222233556···66···610···101515151530···30
size1111555511221···15···52···222222···2

48 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D5D10C3×D5C6×D5
kernelD5×C2×C6C6×D5C2×C30C22×D5D10C2×C10C2×C6C6C22C2
# reps161212226412

Matrix representation of D5×C2×C6 in GL4(𝔽31) generated by

30000
03000
0010
0001
,
5000
03000
0050
0005
,
1000
0100
003014
003013
,
30000
0100
00300
00301
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,30,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,14,13],[30,0,0,0,0,1,0,0,0,0,30,30,0,0,0,1] >;

D5×C2×C6 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_6
% in TeX

G:=Group("D5xC2xC6");
// GroupNames label

G:=SmallGroup(120,44);
// by ID

G=gap.SmallGroup(120,44);
# by ID

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

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

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