Copied to
clipboard

## G = D5×C2×C6order 120 = 23·3·5

### Direct product of C2×C6 and D5

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

 Derived series C1 — C5 — D5×C2×C6
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — D5×C2×C6
 Lower central C5 — D5×C2×C6
 Upper central C1 — C2×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, C2, C3, C22, C22, C5, C6, C6, C23, D5, C10, C2×C6, C2×C6, C15, D10, C2×C10, C22×C6, C3×D5, C30, C22×D5, C6×D5, C2×C30, D5×C2×C6
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, D10, C22×C6, C3×D5, C22×D5, C6×D5, 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 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(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 39 53 43)(2 31 40 54 44)(3 32 41 49 45)(4 33 42 50 46)(5 34 37 51 47)(6 35 38 52 48)(7 13 55 21 30)(8 14 56 22 25)(9 15 57 23 26)(10 16 58 24 27)(11 17 59 19 28)(12 18 60 20 29)
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)(13 58)(14 59)(15 60)(16 55)(17 56)(18 57)(25 28)(26 29)(27 30)(31 51)(32 52)(33 53)(34 54)(35 49)(36 50)(37 40)(38 41)(39 42)

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 kernel D5×C2×C6 C6×D5 C2×C30 C22×D5 D10 C2×C10 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 2 6 4 12

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

 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 14 0 0 30 13
,
 30 0 0 0 0 1 0 0 0 0 30 0 0 0 30 1
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

׿
×
𝔽