C4^2.(C2xC4) - GroupNames

G = C₄².(C₂×C₄) order 128 = 2⁷

2^nd non-split extension by C₄² of C₂×C₄ acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₄.₁₀C₄≀C₂, C₄⋊Q₈.₁C₄, C₁₆⋊C₄.C₂, (C₂×C₈).₂₄D₄, C₄².₂(C₂×C₄), C₄².C₂.₂C₄, C₈⋊C₄.₈₄C₂², C₂².₁₁(C₄.D₄), C₄².₃₀C₂².₂C₂, C₂.₅(C₄².C₂²), (C₂×C₄).₅₇(C₂²⋊C₄), SmallGroup(128,88)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₄² — C₄².(C₂×C₄)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂×C₈ — C₈⋊C₄ — C₄².₃₀C₂² — C₄².(C₂×C₄)

Lower central

C₁ — C₂ — C₂×C₄ — C₄² — C₄².(C₂×C₄)

Upper central

C₁ — C₂ — C₂×C₄ — C₈⋊C₄ — C₄².(C₂×C₄)

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₂×C₄ — C₂×C₄ — C₈⋊C₄ — C₄².(C₂×C₄)

Generators and relations for C₄².(C₂×C₄)
G = < a,b,c,d | a⁴=b⁴=1, c²=b², d⁴=b, ab=ba, cac^-1=a^-1, dad^-1=ab^-1, cbc^-1=b^-1, bd=db, dcd^-1=ab^-1c >

C₁

C₂

₂C₂

C₄

C₂²

C₄

₄C₄

₈C₄

C₂×C₄

₂C₂×C₄

₂C₈

₄Q₈

₄C₂×C₄

₄Q₈

₄C₂×C₄

C₂×C₈

C₄²

₂C₁₆

₂C₄⋊C₄

₂C₂×Q₈

₄C₄⋊C₄

C₄².C₂

C₈⋊C₄

C₄⋊Q₈

₂M₅(2)

₂Q₈⋊C₄

C₄².₃₀C₂²

C₁₆⋊C₄

C₄².(C₂×C₄)

Character table of C₄².(C₂×C₄)

class	1	2A	2B	4A	4B	4C	4D	4E	8A	8B	8C	8D	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	2	2	2	8	16	16	4	4	4	4	8	8	8	8	8	8	8	8
ρ₁	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	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	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	linear of order 2
ρ₅	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	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	-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	-i	i	-i	i	-i	-i	i	i	linear of order 4
ρ₉	2	2	2	2	2	-2	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	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	2	-2	0	0	0	0	0	-2i	2i	0	0	1-i	1+i	-1+i	0	-1-i	0	complex lifted from C₄≀C₂
ρ₁₂	2	2	-2	-2	2	0	0	0	2i	-2i	0	0	1-i	-1-i	0	0	0	-1+i	0	1+i	complex lifted from C₄≀C₂
ρ₁₃	2	2	-2	-2	2	0	0	0	2i	-2i	0	0	-1+i	1+i	0	0	0	1-i	0	-1-i	complex lifted from C₄≀C₂
ρ₁₄	2	2	-2	2	-2	0	0	0	0	0	2i	-2i	0	0	-1-i	-1+i	1+i	0	1-i	0	complex lifted from C₄≀C₂
ρ₁₅	2	2	-2	2	-2	0	0	0	0	0	2i	-2i	0	0	1+i	1-i	-1-i	0	-1+i	0	complex lifted from C₄≀C₂
ρ₁₆	2	2	-2	2	-2	0	0	0	0	0	-2i	2i	0	0	-1+i	-1-i	1-i	0	1+i	0	complex lifted from C₄≀C₂
ρ₁₇	2	2	-2	-2	2	0	0	0	-2i	2i	0	0	-1-i	1-i	0	0	0	1+i	0	-1+i	complex lifted from C₄≀C₂
ρ₁₈	2	2	-2	-2	2	0	0	0	-2i	2i	0	0	1+i	-1+i	0	0	0	-1-i	0	1-i	complex lifted from C₄≀C₂
ρ₁₉	4	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₄.D₄
ρ₂₀	8	-8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₄².(C₂×C₄)
►On 32 points

Generators in S₃₂

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

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

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

(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)

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

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

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

Matrix representation of C₄².(C₂×C₄) ►in GL₈(𝔽₁₇)

7	0	6	11	0	0	0	0
0	7	3	0	0	0	0	0
0	6	10	0	0	0	0	0
14	6	0	10	0	0	0	0
0	0	0	0	7	3	0	11
0	0	0	0	7	10	3	11
0	0	0	0	6	11	10	14
0	0	0	0	3	0	10	7

1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	1	15	0	0	0	0
0	0	1	16	0	0	0	0
0	0	0	0	1	15	0	0
0	0	0	0	1	16	0	0
0	0	0	0	0	0	1	15
0	0	0	0	0	0	1	16

0	0	16	0	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	6	10	0
0	0	0	0	3	0	10	7
0	0	0	0	10	0	11	6
0	0	0	0	10	7	14	6

0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0

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

C₄².(C₂×C₄) in GAP, Magma, Sage, TeX

C_4^2.(C_2\times C_4)

% in TeX

G:=Group("C4^2.(C2xC4)");

// GroupNames label

G:=SmallGroup(128,88);

// by ID

G=gap.SmallGroup(128,88);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,184,1690,521,80,1411,172,4037]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₄².(C₂×C₄) in TeX
Character table of C₄².(C₂×C₄) in TeX

7	0	6	11	0	0	0	0
0	7	3	0	0	0	0	0
0	6	10	0	0	0	0	0
14	6	0	10	0	0	0	0
0	0	0	0	7	3	0	11
0	0	0	0	7	10	3	11
0	0	0	0	6	11	10	14
0	0	0	0	3	0	10	7

1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	1	15	0	0	0	0
0	0	1	16	0	0	0	0
0	0	0	0	1	15	0	0
0	0	0	0	1	16	0	0
0	0	0	0	0	0	1	15
0	0	0	0	0	0	1	16

0	0	16	0	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	6	10	0
0	0	0	0	3	0	10	7
0	0	0	0	10	0	11	6
0	0	0	0	10	7	14	6

0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0

7	0	6	11	0	0	0	0
0	7	3	0	0	0	0	0
0	6	10	0	0	0	0	0
14	6	0	10	0	0	0	0
0	0	0	0	7	3	0	11
0	0	0	0	7	10	3	11
0	0	0	0	6	11	10	14
0	0	0	0	3	0	10	7

1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	1	15	0	0	0	0
0	0	1	16	0	0	0	0
0	0	0	0	1	15	0	0
0	0	0	0	1	16	0	0
0	0	0	0	0	0	1	15
0	0	0	0	0	0	1	16

0	0	16	0	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	6	10	0
0	0	0	0	3	0	10	7
0	0	0	0	10	0	11	6
0	0	0	0	10	7	14	6

0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0

G = C42.(C2×C4) order 128 = 27

2nd non-split extension by C42 of C2×C4 acting faithfully

G = C₄².(C₂×C₄) order 128 = 2⁷

2^nd non-split extension by C₄² of C₂×C₄ acting faithfully

7	0	6	11	0	0	0	0
0	7	3	0	0	0	0	0
0	6	10	0	0	0	0	0
14	6	0	10	0	0	0	0
0	0	0	0	7	3	0	11
0	0	0	0	7	10	3	11
0	0	0	0	6	11	10	14
0	0	0	0	3	0	10	7

1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	1	15	0	0	0	0
0	0	1	16	0	0	0	0
0	0	0	0	1	15	0	0
0	0	0	0	1	16	0	0
0	0	0	0	0	0	1	15
0	0	0	0	0	0	1	16

0	0	16	0	0	0	0	0
0	0	16	1	0	0	0	0
1	0	0	0	0	0	0	0
1	16	0	0	0	0	0	0
0	0	0	0	0	6	10	0
0	0	0	0	3	0	10	7
0	0	0	0	10	0	11	6
0	0	0	0	10	7	14	6

0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0