C4^2.7C2^2 - GroupNames

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L
size	1	1	1	1	4	1	1	1	1	2	2	2	2	4	4	4	2	2	2	2	2	2	2	2	4	4	4	4
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	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	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	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	linear of order 2
ρ₅	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
ρ₆	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
ρ₇	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
ρ₈	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
ρ₉	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	1	1	1	-1	-i	i	i	i	-i	-i	-i	i	-i	i	i	-i	linear of order 4
ρ₁₀	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	1	1	1	-1	i	-i	-i	-i	i	i	i	-i	i	-i	-i	i	linear of order 4
ρ₁₁	1	1	1	1	1	-1	-1	-1	-1	-1	1	-1	1	-1	-1	1	-i	i	i	i	-i	-i	-i	i	i	-i	-i	i	linear of order 4
ρ₁₂	1	1	1	1	1	-1	-1	-1	-1	-1	1	-1	1	-1	-1	1	i	-i	-i	-i	i	i	i	-i	-i	i	i	-i	linear of order 4
ρ₁₃	1	1	1	1	1	-1	-1	-1	-1	1	-1	1	-1	-1	1	-1	-i	-i	i	i	i	i	-i	-i	i	i	-i	-i	linear of order 4
ρ₁₄	1	1	1	1	1	-1	-1	-1	-1	1	-1	1	-1	-1	1	-1	i	i	-i	-i	-i	-i	i	i	-i	-i	i	i	linear of order 4
ρ₁₅	1	1	1	1	-1	-1	-1	-1	-1	1	-1	1	-1	1	-1	1	-i	-i	i	i	i	i	-i	-i	-i	-i	i	i	linear of order 4
ρ₁₆	1	1	1	1	-1	-1	-1	-1	-1	1	-1	1	-1	1	-1	1	i	i	-i	-i	-i	-i	i	i	i	i	-i	-i	linear of order 4
ρ₁₇	2	-2	-2	2	0	-2	-2	2	2	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	0	2	2	-2	-2	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	-2	2	0	-2	-2	2	2	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	0	2	2	-2	-2	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	2	-2	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	2ζ₈	0	0	2ζ₈³	2ζ₈⁷	0	2ζ₈⁵	0	0	0	0	complex lifted from C₈○D₄
ρ₂₂	2	2	-2	-2	0	2i	-2i	2i	-2i	0	0	0	0	0	0	0	2ζ₈⁵	0	2ζ₈⁷	2ζ₈³	0	0	2ζ₈	0	0	0	0	0	complex lifted from C₈○D₄
ρ₂₃	2	2	-2	-2	0	-2i	2i	-2i	2i	0	0	0	0	0	0	0	2ζ₈³	0	2ζ₈	2ζ₈⁵	0	0	2ζ₈⁷	0	0	0	0	0	complex lifted from C₈○D₄
ρ₂₄	2	-2	2	-2	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	2ζ₈³	0	0	2ζ₈	2ζ₈⁵	0	2ζ₈⁷	0	0	0	0	complex lifted from C₈○D₄
ρ₂₅	2	2	-2	-2	0	2i	-2i	2i	-2i	0	0	0	0	0	0	0	2ζ₈	0	2ζ₈³	2ζ₈⁷	0	0	2ζ₈⁵	0	0	0	0	0	complex lifted from C₈○D₄
ρ₂₆	2	-2	2	-2	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	2ζ₈⁷	0	0	2ζ₈⁵	2ζ₈	0	2ζ₈³	0	0	0	0	complex lifted from C₈○D₄
ρ₂₇	2	2	-2	-2	0	-2i	2i	-2i	2i	0	0	0	0	0	0	0	2ζ₈⁷	0	2ζ₈⁵	2ζ₈	0	0	2ζ₈³	0	0	0	0	0	complex lifted from C₈○D₄
ρ₂₈	2	-2	2	-2	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	2ζ₈⁵	0	0	2ζ₈⁷	2ζ₈³	0	2ζ₈	0	0	0	0	complex lifted from C₈○D₄

Smallest permutation representation of C₄².₇C₂²
►On 32 points

Generators in S₃₂

(1 19 27 12)(2 9 28 24)(3 21 29 14)(4 11 30 18)(5 23 31 16)(6 13 32 20)(7 17 25 10)(8 15 26 22)

(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)

(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)

(2 32)(4 26)(6 28)(8 30)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)

G:=sub<Sym(32)| (1,19,27,12)(2,9,28,24)(3,21,29,14)(4,11,30,18)(5,23,31,16)(6,13,32,20)(7,17,25,10)(8,15,26,22), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (2,32)(4,26)(6,28)(8,30)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23)>;

G:=Group( (1,19,27,12)(2,9,28,24)(3,21,29,14)(4,11,30,18)(5,23,31,16)(6,13,32,20)(7,17,25,10)(8,15,26,22), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (2,32)(4,26)(6,28)(8,30)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23) );

G=PermutationGroup([[(1,19,27,12),(2,9,28,24),(3,21,29,14),(4,11,30,18),(5,23,31,16),(6,13,32,20),(7,17,25,10),(8,15,26,22)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(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)], [(2,32),(4,26),(6,28),(8,30),(9,24),(10,14),(11,18),(12,16),(13,20),(15,22),(17,21),(19,23)]])

Matrix representation of C₄².₇C₂² ►in GL₄(𝔽₁₇) generated by

0	16	0	0
1	0	0	0
0	0	0	16
0	0	16	0

4	0	0	0
0	4	0	0
0	0	4	0
0	0	0	4

15	0	0	0
0	15	0	0
0	0	0	8
0	0	9	0

1	0	0	0
0	16	0	0
0	0	1	0
0	0	0	16

G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,0,16,0,0,16,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,15,0,0,0,0,0,9,0,0,8,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C₄².₇C₂² in GAP, Magma, Sage, TeX

C_4^2._7C_2^2

% in TeX

G:=Group("C4^2.7C2^2");

// GroupNames label

G:=SmallGroup(64,114);

// by ID

G=gap.SmallGroup(64,114);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,50,88]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₄².₇C₂² in TeX
Character table of C₄².₇C₂² in TeX

G = C42.7C22 order 64 = 26

7th non-split extension by C42 of C22 acting faithfully

G = C₄².₇C₂² order 64 = 2⁶

7^th non-split extension by C₄² of C₂² acting faithfully