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## G = C2×C52⋊C3order 150 = 2·3·52

### Direct product of C2 and C52⋊C3

Aliases: C2×C52⋊C3, C522C6, (C5×C10)⋊C3, SmallGroup(150,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊C3
 Chief series C1 — C52 — C52⋊C3 — C2×C52⋊C3
 Lower central C52 — C2×C52⋊C3
 Upper central C1 — C2

Generators and relations for C2×C52⋊C3
G = < a,b,c,d | a2=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >

Character table of C2×C52⋊C3

 class 1 2 3A 3B 5A 5B 5C 5D 5E 5F 5G 5H 6A 6B 10A 10B 10C 10D 10E 10F 10G 10H size 1 1 25 25 3 3 3 3 3 3 3 3 25 25 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 -1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ6 ζ65 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ5 1 -1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ65 ζ6 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ7 3 -3 0 0 ζ54+2ζ53 1+√5/2 1-√5/2 1+√5/2 1-√5/2 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 0 0 -2ζ52-ζ5 -ζ53-2ζ5 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -2ζ54-ζ52 -ζ54-2ζ53 complex faithful ρ8 3 3 0 0 ζ54+2ζ53 1+√5/2 1-√5/2 1+√5/2 1-√5/2 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 0 0 2ζ52+ζ5 ζ53+2ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 2ζ54+ζ52 ζ54+2ζ53 complex lifted from C52⋊C3 ρ9 3 -3 0 0 2ζ52+ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 0 0 -ζ54-2ζ53 -2ζ54-ζ52 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -ζ53-2ζ5 -2ζ52-ζ5 complex faithful ρ10 3 3 0 0 2ζ54+ζ52 1-√5/2 1+√5/2 1-√5/2 1+√5/2 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 0 0 ζ53+2ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 1-√5/2 1+√5/2 2ζ52+ζ5 2ζ54+ζ52 complex lifted from C52⋊C3 ρ11 3 -3 0 0 1+√5/2 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 1+√5/2 1-√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 -2ζ54-ζ52 -ζ54-2ζ53 -ζ53-2ζ5 -2ζ52-ζ5 -1+√5/2 -1-√5/2 complex faithful ρ12 3 3 0 0 1-√5/2 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 1-√5/2 1+√5/2 1+√5/2 0 0 1-√5/2 1+√5/2 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 complex lifted from C52⋊C3 ρ13 3 3 0 0 ζ53+2ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 0 0 2ζ54+ζ52 2ζ52+ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 ζ54+2ζ53 ζ53+2ζ5 complex lifted from C52⋊C3 ρ14 3 -3 0 0 2ζ54+ζ52 1-√5/2 1+√5/2 1-√5/2 1+√5/2 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 0 0 -ζ53-2ζ5 -ζ54-2ζ53 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -2ζ52-ζ5 -2ζ54-ζ52 complex faithful ρ15 3 3 0 0 1+√5/2 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 1+√5/2 1-√5/2 1-√5/2 0 0 1+√5/2 1-√5/2 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 complex lifted from C52⋊C3 ρ16 3 3 0 0 1-√5/2 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 1+√5/2 0 0 1-√5/2 1+√5/2 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 complex lifted from C52⋊C3 ρ17 3 -3 0 0 1-√5/2 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 1-√5/2 1+√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 -2ζ52-ζ5 -2ζ54-ζ52 -ζ54-2ζ53 -ζ53-2ζ5 -1-√5/2 -1+√5/2 complex faithful ρ18 3 -3 0 0 1+√5/2 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 1+√5/2 1-√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 -ζ53-2ζ5 -2ζ52-ζ5 -2ζ54-ζ52 -ζ54-2ζ53 -1+√5/2 -1-√5/2 complex faithful ρ19 3 -3 0 0 1-√5/2 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 -ζ54-2ζ53 -ζ53-2ζ5 -2ζ52-ζ5 -2ζ54-ζ52 -1-√5/2 -1+√5/2 complex faithful ρ20 3 3 0 0 2ζ52+ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 0 0 ζ54+2ζ53 2ζ54+ζ52 1+√5/2 1-√5/2 1+√5/2 1-√5/2 ζ53+2ζ5 2ζ52+ζ5 complex lifted from C52⋊C3 ρ21 3 3 0 0 1+√5/2 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 1+√5/2 1-√5/2 1-√5/2 0 0 1+√5/2 1-√5/2 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 complex lifted from C52⋊C3 ρ22 3 -3 0 0 ζ53+2ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 0 0 -2ζ54-ζ52 -2ζ52-ζ5 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -ζ54-2ζ53 -ζ53-2ζ5 complex faithful

Permutation representations of C2×C52⋊C3
On 30 points - transitive group 30T40
Generators in S30
(1 9)(2 8)(3 7)(4 6)(5 10)(11 30)(12 26)(13 27)(14 28)(15 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 4 2 5 3)(6 8 10 7 9)(11 15 14 13 12)(16 18 20 17 19)(21 23 25 22 24)(26 30 29 28 27)
(1 20 26)(2 19 27)(3 18 28)(4 17 29)(5 16 30)(6 21 15)(7 22 14)(8 23 13)(9 24 12)(10 25 11)

G:=sub<Sym(30)| (1,9)(2,8)(3,7)(4,6)(5,10)(11,30)(12,26)(13,27)(14,28)(15,29)(16,25)(17,21)(18,22)(19,23)(20,24), (11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,4,2,5,3)(6,8,10,7,9)(11,15,14,13,12)(16,18,20,17,19)(21,23,25,22,24)(26,30,29,28,27), (1,20,26)(2,19,27)(3,18,28)(4,17,29)(5,16,30)(6,21,15)(7,22,14)(8,23,13)(9,24,12)(10,25,11)>;

G:=Group( (1,9)(2,8)(3,7)(4,6)(5,10)(11,30)(12,26)(13,27)(14,28)(15,29)(16,25)(17,21)(18,22)(19,23)(20,24), (11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,4,2,5,3)(6,8,10,7,9)(11,15,14,13,12)(16,18,20,17,19)(21,23,25,22,24)(26,30,29,28,27), (1,20,26)(2,19,27)(3,18,28)(4,17,29)(5,16,30)(6,21,15)(7,22,14)(8,23,13)(9,24,12)(10,25,11) );

G=PermutationGroup([[(1,9),(2,8),(3,7),(4,6),(5,10),(11,30),(12,26),(13,27),(14,28),(15,29),(16,25),(17,21),(18,22),(19,23),(20,24)], [(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,4,2,5,3),(6,8,10,7,9),(11,15,14,13,12),(16,18,20,17,19),(21,23,25,22,24),(26,30,29,28,27)], [(1,20,26),(2,19,27),(3,18,28),(4,17,29),(5,16,30),(6,21,15),(7,22,14),(8,23,13),(9,24,12),(10,25,11)]])

G:=TransitiveGroup(30,40);

C2×C52⋊C3 is a maximal subgroup of   C522Dic3  C522C12

Matrix representation of C2×C52⋊C3 in GL3(𝔽11) generated by

 10 0 0 0 10 0 0 0 10
,
 2 5 2 3 0 6 6 1 2
,
 6 5 4 10 7 9 0 0 4
,
 1 0 1 0 0 10 0 1 10
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[2,3,6,5,0,1,2,6,2],[6,10,0,5,7,0,4,9,4],[1,0,0,0,0,1,1,10,10] >;

C2×C52⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes C_3
% in TeX

G:=Group("C2xC5^2:C3");
// GroupNames label

G:=SmallGroup(150,7);
// by ID

G=gap.SmallGroup(150,7);
# by ID

G:=PCGroup([4,-2,-3,-5,5,582,919]);
// Polycyclic

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

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