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G = C2×C52⋊C6  order 300 = 22·3·52

Direct product of C2 and C52⋊C6

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×C52⋊C6, C5⋊D5⋊C6, (C5×C10)⋊C6, C52⋊(C2×C6), C52⋊C3⋊3C22, (C2×C5⋊D5)⋊C3, (C2×C52⋊C3)⋊2C2, SmallGroup(300,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C2×C52⋊C6
Chief series
C1 — C52 — C52⋊C3 — C52⋊C6 — C2×C52⋊C6
Lower central
C52 — C2×C52⋊C6
Upper central
C1 — C2

Generators and relations for C2×C52⋊C6
 G = < a,b,c,d | a2=b5=c5=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2c3, dcd-1=b-1c-1 >

C1
C2
25C2
25C2
25C3
3C5
3C5
25C22
25C6
25C6
25C6
3C10
3C10
15D5
15D5
15D5
15D5
C52
25C2×C6
15D10
15D10
C5⋊D5
C5×C10
C5⋊D5
C52⋊C3
C2×C5⋊D5
C52⋊C6
C2×C52⋊C3
C52⋊C6
C2×C52⋊C6

Character table of C2×C52⋊C6

 class 12A2B2C3A3B5A5B5C5D6A6B6C6D6E6F10A10B10C10D
 size 112525252566662525252525256666
ρ111111111111111111111    trivial
ρ211-1-1111111-1-1-11-111111    linear of order 2
ρ31-1-11111111-111-1-1-1-1-1-1-1    linear of order 2
ρ41-11-11111111-1-1-11-1-1-1-1-1    linear of order 2
ρ51111ζ3ζ321111ζ3ζ32ζ3ζ3ζ32ζ321111    linear of order 3
ρ611-1-1ζ32ζ31111ζ6ζ65ζ6ζ32ζ65ζ31111    linear of order 6
ρ71111ζ32ζ31111ζ32ζ3ζ32ζ32ζ3ζ31111    linear of order 3
ρ81-1-11ζ3ζ321111ζ65ζ32ζ3ζ65ζ6ζ6-1-1-1-1    linear of order 6
ρ91-11-1ζ3ζ321111ζ3ζ6ζ65ζ65ζ32ζ6-1-1-1-1    linear of order 6
ρ101-1-11ζ32ζ31111ζ6ζ3ζ32ζ6ζ65ζ65-1-1-1-1    linear of order 6
ρ111-11-1ζ32ζ31111ζ32ζ65ζ6ζ6ζ3ζ65-1-1-1-1    linear of order 6
ρ1211-1-1ζ3ζ321111ζ65ζ6ζ65ζ3ζ6ζ321111    linear of order 6
ρ136-600001+√5-3-√5/2-3+√5/21-√50000003+√5/23-√5/2-1+√5-1-√5    orthogonal faithful
ρ146600001+√5-3-√5/2-3+√5/21-√5000000-3-√5/2-3+√5/21-√51+√5    orthogonal lifted from C52⋊C6
ρ156600001-√5-3+√5/2-3-√5/21+√5000000-3+√5/2-3-√5/21+√51-√5    orthogonal lifted from C52⋊C6
ρ16660000-3-√5/21-√51+√5-3+√5/20000001-√51+√5-3+√5/2-3-√5/2    orthogonal lifted from C52⋊C6
ρ176-600001-√5-3+√5/2-3-√5/21+√50000003-√5/23+√5/2-1-√5-1+√5    orthogonal faithful
ρ186-60000-3-√5/21-√51+√5-3+√5/2000000-1+√5-1-√53-√5/23+√5/2    orthogonal faithful
ρ196-60000-3+√5/21+√51-√5-3-√5/2000000-1-√5-1+√53+√5/23-√5/2    orthogonal faithful
ρ20660000-3+√5/21+√51-√5-3-√5/20000001+√51-√5-3-√5/2-3+√5/2    orthogonal lifted from C52⋊C6

Permutation representations of C2×C52⋊C6
►On 30 points - transitive group 30T68
Generators in S30
(1 6)(2 4)(3 5)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)

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

(1 14 12 9 17)(2 15 7 10 18)(4 19 25 28 22)(6 24 30 27 21)

(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)
 
G:=sub<Sym(30)| (1,6)(2,4)(3,5)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,9,14,17,12)(2,18,10,7,15)(3,16,8,11,13)(4,22,28,25,19)(5,20,26,29,23)(6,27,24,21,30), (1,14,12,9,17)(2,15,7,10,18)(4,19,25,28,22)(6,24,30,27,21), (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)>;
 
G:=Group( (1,6)(2,4)(3,5)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,9,14,17,12)(2,18,10,7,15)(3,16,8,11,13)(4,22,28,25,19)(5,20,26,29,23)(6,27,24,21,30), (1,14,12,9,17)(2,15,7,10,18)(4,19,25,28,22)(6,24,30,27,21), (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) );
 
G=PermutationGroup([[(1,6),(2,4),(3,5),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22)], [(1,9,14,17,12),(2,18,10,7,15),(3,16,8,11,13),(4,22,28,25,19),(5,20,26,29,23),(6,27,24,21,30)], [(1,14,12,9,17),(2,15,7,10,18),(4,19,25,28,22),(6,24,30,27,21)], [(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)]])
 
G:=TransitiveGroup(30,68);
 

Matrix representation of C2×C52⋊C6 ►in GL6(𝔽31)

3000000
0300000
0030000
0003000
0000300
0000030
,
19300000
20300000
00121800
0012000
00001930
00002030
,
3010000
11190000
00193000
00203000
000010
000001
,
00193000
00191200
00001930
00001912
19300000
19120000

G:=sub<GL(6,GF(31))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[19,20,0,0,0,0,30,30,0,0,0,0,0,0,12,12,0,0,0,0,18,0,0,0,0,0,0,0,19,20,0,0,0,0,30,30],[30,11,0,0,0,0,1,19,0,0,0,0,0,0,19,20,0,0,0,0,30,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,0,0,19,19,0,0,0,0,30,12,19,19,0,0,0,0,30,12,0,0,0,0,0,0,19,19,0,0,0,0,30,12,0,0] >;
 

C2×C52⋊C6 in GAP, Magma, Sage, TeX 

C_2\times C_5^2\rtimes C_6
 
% in TeX
 
G:=Group("C2xC5^2:C6");
 
// GroupNames label
 
G:=SmallGroup(300,27);
 
// by ID
 
G=gap.SmallGroup(300,27);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-5,5,963,793,6004,464]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^5=c^5=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2*c^3,d*c*d^-1=b^-1*c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C52⋊C6 in TeX
Character table of C2×C52⋊C6 in TeX

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