C2xC3^2:2C8 - GroupNames

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	4	4	9	9	9	9	4	4	4	4	4	4	9	9	9	9	9	9	9	9
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	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	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	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	linear of order 2
ρ₅	1	1	1	1	1	1	-1	-1	-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	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	-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	-1	1	1	-1	-1	-1	i	i	-i	-i	-i	-i	i	i	linear of order 4
ρ₉	1	-1	1	-1	1	1	i	i	-i	-i	-1	-1	-1	-1	1	1	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	linear of order 8
ρ₁₀	1	-1	1	-1	1	1	i	i	-i	-i	-1	-1	-1	-1	1	1	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	linear of order 8
ρ₁₁	1	1	-1	-1	1	1	i	-i	i	-i	1	-1	-1	1	-1	-1	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	linear of order 8
ρ₁₂	1	1	-1	-1	1	1	i	-i	i	-i	1	-1	-1	1	-1	-1	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	linear of order 8
ρ₁₃	1	-1	1	-1	1	1	-i	-i	i	i	-1	-1	-1	-1	1	1	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	linear of order 8
ρ₁₄	1	1	-1	-1	1	1	-i	i	-i	i	1	-1	-1	1	-1	-1	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	linear of order 8
ρ₁₅	1	1	-1	-1	1	1	-i	i	-i	i	1	-1	-1	1	-1	-1	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	linear of order 8
ρ₁₆	1	-1	1	-1	1	1	-i	-i	i	i	-1	-1	-1	-1	1	1	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	linear of order 8
ρ₁₇	4	4	4	4	-2	1	0	0	0	0	1	-2	1	-2	1	-2	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₁₈	4	-4	-4	4	1	-2	0	0	0	0	2	1	-2	-1	2	-1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₃²⋊C₄
ρ₁₉	4	4	4	4	1	-2	0	0	0	0	-2	1	-2	1	-2	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₂₀	4	-4	-4	4	-2	1	0	0	0	0	-1	-2	1	2	-1	2	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₃²⋊C₄
ρ₂₁	4	-4	4	-4	1	-2	0	0	0	0	2	-1	2	-1	-2	1	0	0	0	0	0	0	0	0	symplectic lifted from C₃²⋊₂C₈, Schur index 2
ρ₂₂	4	4	-4	-4	-2	1	0	0	0	0	1	2	-1	-2	-1	2	0	0	0	0	0	0	0	0	symplectic lifted from C₃²⋊₂C₈, Schur index 2
ρ₂₃	4	-4	4	-4	-2	1	0	0	0	0	-1	2	-1	2	1	-2	0	0	0	0	0	0	0	0	symplectic lifted from C₃²⋊₂C₈, Schur index 2
ρ₂₄	4	4	-4	-4	1	-2	0	0	0	0	-2	-1	2	1	2	-1	0	0	0	0	0	0	0	0	symplectic lifted from C₃²⋊₂C₈, Schur index 2

Smallest permutation representation of C₂×C₃²⋊₂C₈
►On 48 points

Generators in S₄₈

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

(2 32 22)(4 24 26)(6 28 18)(8 20 30)(10 36 42)(12 44 38)(14 40 46)(16 48 34)

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

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

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

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

C₂×C₃²⋊₂C₈ is a maximal subgroup of
C₆².₃D₄ C₆².₄D₄ C₆².₆D₄ C₆².₇D₄ C₆².₂Q₈ (C₃×C₁₂)⋊₄C₈ C₃²⋊₂C₈⋊C₄ C₆².₆(C₂×C₄) C₃²⋊₅(C₄⋊C₈) C₆²⋊₃C₈ C₂².F₉ C₆².₁₃D₄ C₆².(C₂×C₄)
C₂×C₃²⋊₂C₈ is a maximal quotient of
C₆².₄C₈ (C₃×C₁₂)⋊₄C₈ C₆²⋊₃C₈

Matrix representation of C₂×C₃²⋊₂C₈ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	72	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	72	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	72	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	72	72

51	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	41	51	0	0
0	0	10	32	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[51,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,41,10,0,0,0,0,51,32,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂×C₃²⋊₂C₈ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2C_8

% in TeX

G:=Group("C2xC3^2:2C8");

// GroupNames label

G:=SmallGroup(144,134);

// by ID

G=gap.SmallGroup(144,134);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,50,3364,256,4613,881]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;

// generators/relations

Export

Subgroup lattice of C₂×C₃²⋊₂C₈ in TeX
Character table of C₂×C₃²⋊₂C₈ in TeX

G = C2×C32⋊2C8 order 144 = 24·32

Direct product of C2 and C32⋊2C8

G = C₂×C₃²⋊₂C₈ order 144 = 2⁴·3²

Direct product of C₂ and C₃²⋊₂C₈