C2^2.C4^2 - GroupNames

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	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	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	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	linear of order 2
ρ₅	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	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	-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	-i	i	-i	1	i	linear of order 4
ρ₉	1	-1	1	-1	-1	1	-1	1	-1	1	-i	i	-i	i	-1	1	-1	i	-i	i	1	-i	linear of order 4
ρ₁₀	1	-1	1	-1	-1	1	1	-1	1	-1	i	-i	-i	i	-i	-i	i	1	1	-1	i	-1	linear of order 4
ρ₁₁	1	-1	1	-1	-1	1	1	-1	1	-1	-i	i	i	-i	-i	-i	i	-1	-1	1	i	1	linear of order 4
ρ₁₂	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
ρ₁₃	1	-1	1	-1	-1	1	-1	1	-1	1	-i	i	-i	i	1	-1	1	-i	i	-i	-1	i	linear of order 4
ρ₁₄	1	-1	1	-1	-1	1	-1	1	-1	1	i	-i	i	-i	1	-1	1	i	-i	i	-1	-i	linear of order 4
ρ₁₅	1	-1	1	-1	-1	1	1	-1	1	-1	-i	i	i	-i	i	i	-i	1	1	-1	-i	-1	linear of order 4
ρ₁₆	1	-1	1	-1	-1	1	1	-1	1	-1	i	-i	-i	i	i	i	-i	-1	-1	1	-i	1	linear of order 4
ρ₁₇	2	2	2	2	-2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	2	-2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	2	-2	2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₁	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₄.D₄
ρ₂₂	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄.₁₀D₄, Schur index 2

Smallest permutation representation of C₂².C₄²
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

C₂².C₄² is a maximal subgroup of
C₂.C₂≀C₄ (C₂×C₄).D₈ (C₂×C₄).Q₁₆ C₂.₇C₂≀C₄ C₂³.₁₅C₄² C₄×C₄.D₄ C₄×C₄.₁₀D₄ C₄².₉₆D₄ C₄².₉₇D₄ C₄○D₄.₄Q₈ C₄○D₄.₅Q₈ (C₂²×C₄).₂₇₅D₄ (C₂²×C₄).₂₇₆D₄ C₄≀C₂⋊C₄ C₄²⋊₉(C₂×C₄) C₈.C₂²⋊C₄ C₈⋊C₂²⋊C₄ M₄(2)⋊₂₀D₄ M₄(2).₄₅D₄ M₄(2).₄₈D₄ M₄(2).₄₉D₄ C₄.₁₀D₄⋊₃C₄ C₄.D₄⋊₃C₄ M₄(2)⋊₈Q₈ C₄².₁₂₈D₄ M₄(2)⋊D₄ M₄(2)⋊₄D₄ (C₂×D₄)⋊₂Q₈ (C₂×Q₈)⋊₂Q₈ C₄²⋊₁₁D₄ C₄²⋊₁₂D₄ M₄(2).₁₀D₄ M₄(2).₁₁D₄ M₄(2)⋊Q₈ C₄²⋊₃Q₈ M₄(2).₁₂D₄ M₄(2).₁₃D₄ (C₂×C₈).₅₅D₄ (C₂×C₈).₁₆₅D₄ M₄(2).Q₈ M₄(2).₂Q₈ C₂².F₅⋊C₄
C_4p.(C₄⋊C₄): M₄(2).₅Q₈ M₄(2).₆Q₈ C₁₂.(C₄⋊C₄) M₄(2)⋊Dic₃ (C₂×C₂₀).Q₈ M₄(2)⋊Dic₅ M₄(2)⋊F₅ C₂₈.(C₄⋊C₄) ...
C₂².C₄² is a maximal quotient of
C₂³.₁₉C₄² C₄².₃Q₈ C₄².₄Q₈ C₄².₂₅D₄ C₄².₇Q₈ C₄².₃₈₈D₄ C₂³.C₄² C₄².₃₀D₄ C₄².₃₂D₄ M₄(2)⋊F₅ C₂².F₅⋊C₄
(C₂×C₄).D_4p: C₄².₂₇D₄ C₄².₈Q₈ C₁₂.(C₄⋊C₄) M₄(2)⋊Dic₃ (C₂×C₂₀).Q₈ M₄(2)⋊Dic₅ C₂₈.(C₄⋊C₄) M₄(2)⋊Dic₇ ...

Matrix representation of C₂².C₄² ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	11	3	16	0
0	0	6	14	0	16

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	11	3	15	0
0	0	0	0	1	1
0	0	10	9	16	10
0	0	8	9	1	7

4	0	0	0	0	0
0	13	0	0	0	0
0	0	10	12	0	0
0	0	13	7	0	0
0	0	11	2	4	11
0	0	13	3	11	13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,11,6,0,0,0,1,3,14,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,0,10,8,0,0,3,0,9,9,0,0,15,1,16,1,0,0,0,1,10,7],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,10,13,11,13,0,0,12,7,2,3,0,0,0,0,4,11,0,0,0,0,11,13] >;

C₂².C₄² in GAP, Magma, Sage, TeX

C_2^2.C_4^2

% in TeX

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

// GroupNames label

G:=SmallGroup(64,24);

// by ID

G=gap.SmallGroup(64,24);

# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489,117]);

// Polycyclic

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

// generators/relations

Export

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

G = C22.C42 order 64 = 26

2nd non-split extension by C22 of C42 acting via C42/C2×C4=C2

G = C₂².C₄² order 64 = 2⁶

2^nd non-split extension by C₂² of C₄² acting via C₄²/C₂×C₄=C₂