C4.19S3wrC2 - GroupNames

G = C₄.₁₉S₃≀C₂ order 288 = 2⁵·3²

4^th central extension by C₄ of S₃≀C₂

Aliases: C₄.₁₉S₃≀C₂, (C₃×C₁₂).₁₉D₄, C₃²⋊₂C₈.₄C₄, C₃²⋊₁(C₈.C₄), C₁₂.₃₁D₆.₂C₂, (C₂×C₃⋊S₃).₁Q₈, (C₃×C₆).₂(C₄⋊C₄), C₃⋊S₃⋊₃C₈.₃C₂, C₃⋊Dic₃.₈(C₂×C₄), C₂.₃(C₃⋊S₃.Q₈), (C₄×C₃⋊S₃).₅₃C₂², SmallGroup(288,381)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊Dic₃ — C₄.₁₉S₃≀C₂

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₄×C₃⋊S₃ — C₁₂.₃₁D₆ — C₄.₁₉S₃≀C₂

Lower central

C₃² — C₃×C₆ — C₃⋊Dic₃ — C₄.₁₉S₃≀C₂

Upper central

C₁ — C₄

Generators and relations for C₄.₁₉S₃≀C₂
G = < a,b,c,d,e | a⁴=b³=c³=1, d⁴=a², e²=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=c, ebe^-1=dcd^-1=b^-1, ce=ec, ede^-1=d³ >

C₁

C₂

₁₈C₂

₂C₃

C₄

₉C₄

₉C₂²

₂C₆

₁₂S₃

C₃²

₆C₈

₉C₂×C₄

₉C₈

₂C₁₂

₆Dic₃

₆D₆

C₃×C₆

₂C₃⋊S₃

₉M₄(2)

₉C₂×C₈

₉M₄(2)

₂C₃⋊C₈

₆C₂₄

₆C₄×S₃

C₃×C₁₂

C₃⋊Dic₃

C₂×C₃⋊S₃

₉C₈.C₄

₆C₈⋊S₃

C₃²⋊₂C₈

C₄×C₃⋊S₃

C₃²⋊₂C₈

₂C₃×C₃⋊C₈

C₁₂.₃₁D₆

C₃⋊S₃⋊₃C₈

C₄.₁₉S₃≀C₂

Character table of C₄.₁₉S₃≀C₂

class	1	2A	2B	3A	3B	4A	4B	4C	6A	6B	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	12C	12D	24A	24B	24C	24D	24E	24F	24G	24H
size	1	1	18	4	4	1	1	18	4	4	12	12	12	12	18	18	18	18	4	4	4	4	12	12	12	12	12	12	12	12
ρ₁	1	1	1	1	1	1	1	1	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	1	-1	-1	1	1	-1	-1	1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	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
ρ₄	1	1	1	1	1	1	1	1	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
ρ₅	1	1	-1	1	1	-1	-1	1	1	1	i	i	-i	-i	1	-1	1	-1	-1	-1	-1	-1	i	i	i	i	-i	-i	-i	-i	linear of order 4
ρ₆	1	1	-1	1	1	-1	-1	1	1	1	i	-i	-i	i	-1	1	-1	1	-1	-1	-1	-1	i	-i	-i	i	-i	i	i	-i	linear of order 4
ρ₇	1	1	-1	1	1	-1	-1	1	1	1	-i	-i	i	i	1	-1	1	-1	-1	-1	-1	-1	-i	-i	-i	-i	i	i	i	i	linear of order 4
ρ₈	1	1	-1	1	1	-1	-1	1	1	1	-i	i	i	-i	-1	1	-1	1	-1	-1	-1	-1	-i	i	i	-i	i	-i	-i	i	linear of order 4
ρ₉	2	2	-2	2	2	2	2	-2	2	2	0	0	0	0	0	0	0	0	2	2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₁	2	-2	0	2	2	2i	-2i	0	-2	-2	0	0	0	0	√2	√-2	-√2	-√-2	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₈.C₄
ρ₁₂	2	-2	0	2	2	-2i	2i	0	-2	-2	0	0	0	0	√2	-√-2	-√2	√-2	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₈.C₄
ρ₁₃	2	-2	0	2	2	-2i	2i	0	-2	-2	0	0	0	0	-√2	√-2	√2	-√-2	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₈.C₄
ρ₁₄	2	-2	0	2	2	2i	-2i	0	-2	-2	0	0	0	0	-√2	-√-2	√2	√-2	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₈.C₄
ρ₁₅	4	4	0	-2	1	4	4	0	1	-2	0	-2	0	-2	0	0	0	0	1	1	-2	-2	0	1	1	0	0	1	1	0	orthogonal lifted from S₃≀C₂
ρ₁₆	4	4	0	1	-2	4	4	0	-2	1	-2	0	-2	0	0	0	0	0	-2	-2	1	1	1	0	0	1	1	0	0	1	orthogonal lifted from S₃≀C₂
ρ₁₇	4	4	0	-2	1	4	4	0	1	-2	0	2	0	2	0	0	0	0	1	1	-2	-2	0	-1	-1	0	0	-1	-1	0	orthogonal lifted from S₃≀C₂
ρ₁₈	4	4	0	1	-2	4	4	0	-2	1	2	0	2	0	0	0	0	0	-2	-2	1	1	-1	0	0	-1	-1	0	0	-1	orthogonal lifted from S₃≀C₂
ρ₁₉	4	4	0	-2	1	-4	-4	0	1	-2	0	-2i	0	2i	0	0	0	0	-1	-1	2	2	0	i	i	0	0	-i	-i	0	complex lifted from C₃⋊S₃.Q₈
ρ₂₀	4	4	0	1	-2	-4	-4	0	-2	1	-2i	0	2i	0	0	0	0	0	2	2	-1	-1	i	0	0	i	-i	0	0	-i	complex lifted from C₃⋊S₃.Q₈
ρ₂₁	4	4	0	1	-2	-4	-4	0	-2	1	2i	0	-2i	0	0	0	0	0	2	2	-1	-1	-i	0	0	-i	i	0	0	i	complex lifted from C₃⋊S₃.Q₈
ρ₂₂	4	4	0	-2	1	-4	-4	0	1	-2	0	2i	0	-2i	0	0	0	0	-1	-1	2	2	0	-i	-i	0	0	i	i	0	complex lifted from C₃⋊S₃.Q₈
ρ₂₃	4	-4	0	-2	1	4i	-4i	0	-1	2	0	0	0	0	0	0	0	0	i	-i	2i	-2i	0	2ζ₈ζ₃+ζ₈	2ζ₈⁵ζ₃+ζ₈⁵	0	0	2ζ₈⁷ζ₃+ζ₈⁷	2ζ₈³ζ₃+ζ₈³	0	complex faithful
ρ₂₄	4	-4	0	-2	1	-4i	4i	0	-1	2	0	0	0	0	0	0	0	0	-i	i	-2i	2i	0	2ζ₈³ζ₃+ζ₈³	2ζ₈⁷ζ₃+ζ₈⁷	0	0	2ζ₈⁵ζ₃+ζ₈⁵	2ζ₈ζ₃+ζ₈	0	complex faithful
ρ₂₅	4	-4	0	-2	1	-4i	4i	0	-1	2	0	0	0	0	0	0	0	0	-i	i	-2i	2i	0	2ζ₈⁷ζ₃+ζ₈⁷	2ζ₈³ζ₃+ζ₈³	0	0	2ζ₈ζ₃+ζ₈	2ζ₈⁵ζ₃+ζ₈⁵	0	complex faithful
ρ₂₆	4	-4	0	1	-2	4i	-4i	0	2	-1	0	0	0	0	0	0	0	0	-2i	2i	-i	i	2ζ₈⁵ζ₃+ζ₈⁵	0	0	2ζ₈ζ₃+ζ₈	2ζ₈⁷ζ₃+ζ₈⁷	0	0	2ζ₈³ζ₃+ζ₈³	complex faithful
ρ₂₇	4	-4	0	1	-2	-4i	4i	0	2	-1	0	0	0	0	0	0	0	0	2i	-2i	i	-i	2ζ₈⁷ζ₃+ζ₈⁷	0	0	2ζ₈³ζ₃+ζ₈³	2ζ₈⁵ζ₃+ζ₈⁵	0	0	2ζ₈ζ₃+ζ₈	complex faithful
ρ₂₈	4	-4	0	1	-2	-4i	4i	0	2	-1	0	0	0	0	0	0	0	0	2i	-2i	i	-i	2ζ₈³ζ₃+ζ₈³	0	0	2ζ₈⁷ζ₃+ζ₈⁷	2ζ₈ζ₃+ζ₈	0	0	2ζ₈⁵ζ₃+ζ₈⁵	complex faithful
ρ₂₉	4	-4	0	1	-2	4i	-4i	0	2	-1	0	0	0	0	0	0	0	0	-2i	2i	-i	i	2ζ₈ζ₃+ζ₈	0	0	2ζ₈⁵ζ₃+ζ₈⁵	2ζ₈³ζ₃+ζ₈³	0	0	2ζ₈⁷ζ₃+ζ₈⁷	complex faithful
ρ₃₀	4	-4	0	-2	1	4i	-4i	0	-1	2	0	0	0	0	0	0	0	0	i	-i	2i	-2i	0	2ζ₈⁵ζ₃+ζ₈⁵	2ζ₈ζ₃+ζ₈	0	0	2ζ₈³ζ₃+ζ₈³	2ζ₈⁷ζ₃+ζ₈⁷	0	complex faithful

Smallest permutation representation of C₄.₁₉S₃≀C₂
►On 48 points

Generators in S₄₈

(1 7 5 3)(2 8 6 4)(9 37 13 33)(10 38 14 34)(11 39 15 35)(12 40 16 36)(17 19 21 23)(18 20 22 24)(25 43 29 47)(26 44 30 48)(27 45 31 41)(28 46 32 42)

(2 47 27)(4 29 41)(6 43 31)(8 25 45)(9 18 35)(11 37 20)(13 22 39)(15 33 24)

(1 46 26)(3 28 48)(5 42 30)(7 32 44)(10 36 19)(12 21 38)(14 40 23)(16 17 34)

(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)

(1 23 7 17 5 19 3 21)(2 18 8 20 6 22 4 24)(9 25 37 43 13 29 33 47)(10 28 38 46 14 32 34 42)(11 31 39 41 15 27 35 45)(12 26 40 44 16 30 36 48)

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

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

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

Matrix representation of C₄.₁₉S₃≀C₂ ►in GL₄(𝔽₅) generated by

3	0	0	0
0	3	0	0
0	0	3	0
0	0	0	3

2	0	2	0
0	2	0	1
4	0	2	0
0	3	0	2

2	0	2	0
0	2	0	4
4	0	2	0
0	2	0	2

0	0	0	3
0	0	4	0
0	3	0	0
1	0	0	0

0	3	0	0
1	0	0	0
0	0	0	1
0	0	3	0

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

C₄.₁₉S₃≀C₂ in GAP, Magma, Sage, TeX

C_4._{19}S_3\wr C_2

% in TeX

G:=Group("C4.19S3wrC2");

// GroupNames label

G:=SmallGroup(288,381);

// by ID

G=gap.SmallGroup(288,381);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,422,219,100,80,2693,2028,691,797,2372]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^3=1,d^4=a^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e^-1=d*c*d^-1=b^-1,c*e=e*c,e*d*e^-1=d^3>;

// generators/relations

Export

Subgroup lattice of C₄.₁₉S₃≀C₂ in TeX
Character table of C₄.₁₉S₃≀C₂ in TeX

G = C4.19S3≀C2 order 288 = 25·32

4th central extension by C4 of S3≀C2

G = C₄.₁₉S₃≀C₂ order 288 = 2⁵·3²

4^th central extension by C₄ of S₃≀C₂