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## 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

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