Copied to
clipboard

G = C2×C32⋊Dic5order 360 = 23·32·5

Direct product of C2 and C32⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C2×C32⋊Dic5
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — C32⋊Dic5 — C2×C32⋊Dic5
 Lower central C3×C15 — C2×C32⋊Dic5
 Upper central C1 — C2

Generators and relations for C2×C32⋊Dic5
G = < a,b,c,d,e | a2=b3=c3=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, ebe-1=bc-1, dcd-1=c-1, ece-1=b-1c-1, ede-1=d-1 >

Smallest permutation representation of C2×C32⋊Dic5
On 60 points
Generators in S60
(1 6)(2 7)(3 8)(4 9)(5 10)(11 18)(12 19)(13 20)(14 16)(15 17)(21 58)(22 59)(23 60)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 40 35)(2 36 31)(3 32 37)(4 38 33)(5 34 39)(6 48 43)(7 44 49)(8 50 45)(9 46 41)(10 42 47)(11 55 60)(12 51 56)(13 57 52)(14 53 58)(15 59 54)(16 26 21)(17 22 27)(18 28 23)(19 24 29)(20 30 25)
(1 35 40)(2 31 36)(3 37 32)(4 33 38)(5 39 34)(6 43 48)(7 49 44)(8 45 50)(9 41 46)(10 47 42)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 13)(7 12)(8 11)(9 15)(10 14)(21 39 26 34)(22 38 27 33)(23 37 28 32)(24 36 29 31)(25 35 30 40)(41 59 46 54)(42 58 47 53)(43 57 48 52)(44 56 49 51)(45 55 50 60)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 10A 10B 10C 10D 10E 10F 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 4 4 4 4 5 5 6 6 10 10 10 10 10 10 15 ··· 15 30 ··· 30 size 1 1 9 9 4 4 45 45 45 45 2 2 4 4 2 2 18 18 18 18 4 ··· 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C4 C4 D5 Dic5 D10 Dic5 C32⋊C4 C2×C32⋊C4 C32⋊Dic5 C2×C32⋊Dic5 kernel C2×C32⋊Dic5 C32⋊Dic5 C10×C3⋊S3 C5×C3⋊S3 C3×C30 C2×C3⋊S3 C3⋊S3 C3⋊S3 C3×C6 C10 C5 C2 C1 # reps 1 2 1 2 2 2 2 2 2 2 2 8 8

Matrix representation of C2×C32⋊Dic5 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 0 60 60 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 60
,
 0 1 0 0 0 0 60 18 0 0 0 0 0 0 58 0 0 0 0 0 3 3 0 0 0 0 0 0 20 0 0 0 0 0 41 41
,
 0 11 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 60 60 0 0

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[0,60,0,0,0,0,1,18,0,0,0,0,0,0,58,3,0,0,0,0,0,3,0,0,0,0,0,0,20,41,0,0,0,0,0,41],[0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C32⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xC3^2:Dic5");
// GroupNames label

G:=SmallGroup(360,149);
// by ID

G=gap.SmallGroup(360,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,963,111,964,376,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,e*b*e^-1=b*c^-1,d*c*d^-1=c^-1,e*c*e^-1=b^-1*c^-1,e*d*e^-1=d^-1>;
// generators/relations

Export

׿
×
𝔽