C4.(C4xD4) - GroupNames

G = C₄.(C₄×D₄) order 128 = 2⁷

5^th non-split extension by C₄ of C₄×D₄ acting via C₄×D₄/C₄²=C₂

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄⋊Q₈⋊₃C₄, C₄.₅(C₄×D₄), (C₂×C₈).₃₀D₄, (C₂×D₄).₇₉D₄, C₄.₄D₄⋊₁C₄, C₄².₇(C₂×C₄), C₄.₉C₄²⋊₉C₂, C₄.₉₉C₂²≀C₂, (C₂²×C₄).₆₇D₄, C₂³.₁₃₂(C₂×D₄), Q₈○M₄(2).₅C₂, (C₂²×C₄).₃₃C₂³, C₂³.₃₈D₄⋊₂₅C₂, C₂².₅₃(C₄⋊D₄), C₂³.₁₃(C₂²⋊C₄), (C₂²×Q₈).₂₄C₂², C₄²⋊C₂.₃₁C₂², C₄.₁₃(C₂².D₄), C₂³.C₂³.₇C₂, C₂.₄₈(C₂³.₂₃D₄), (C₂×M₄(2)).₁₉₂C₂², C₂³.₃₈C₂³.₁C₂, (C₂×D₄).₈₈(C₂×C₄), (C₂×C₄).₂₄₄(C₂×D₄), (C₂×Q₈).₇₆(C₂×C₄), (C₂×C₄).₃₂₈(C₄○D₄), (C₂×C₄).₁₇(C₂²⋊C₄), (C₂×C₄).₁₉₃(C₂²×C₄), (C₂×C₄○D₄).₂₇C₂², C₂².₄₇(C₂×C₂²⋊C₄), SmallGroup(128,641)

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

Derived series

C₁ — C₂×C₄ — C₄.(C₄×D₄)

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄○D₄ — Q₈○M₄(2) — C₄.(C₄×D₄)

Lower central

C₁ — C₂ — C₂×C₄ — C₄.(C₄×D₄)

Upper central

C₁ — C₂ — C₂²×C₄ — C₄.(C₄×D₄)

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₄.(C₄×D₄)

Generators and relations for C₄.(C₄×D₄)
G = < a,b,c,d | a⁴=c⁴=1, b⁴=d²=a², ab=ba, ac=ca, dad^-1=a^-1, cbc^-1=a^-1b, bd=db, dcd^-1=c^-1 >

Subgroups: 300 in 156 conjugacy classes, 54 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂³⋊C₄, Q₈⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊Q₈, C₂×M₄(2), C₂×M₄(2), C₈○D₄, C₂²×Q₈, C₂×C₄○D₄, C₄.₉C₄², C₂³.C₂³, C₂³.₃₈D₄, C₂³.₃₈C₂³, Q₈○M₄(2), C₄.(C₄×D₄)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂³.₂₃D₄, C₄.(C₄×D₄)

Character table of C₄.(C₄×D₄)

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	2	2	2	4	4	2	2	2	2	4	4	8	8	8	8	8	8	8	8	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	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	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	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	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	-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	-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	-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	-1	linear of order 2
ρ₉	1	1	-1	-1	1	1	1	-1	1	-1	1	-1	-1	1	i	i	1	-i	-i	-1	-1	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	i	-i	-1	-i	i	1	-1	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	-i	i	1	i	-i	-1	1	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	-i	-i	-1	i	i	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	1	1	1	-1	i	-i	1	-i	i	-1	1	-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	i	i	-1	-i	-i	1	1	-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	-i	-i	1	i	i	-1	-1	-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	-i	i	-1	i	-i	1	-1	-i	i	-i	-i	i	-i	i	i	linear of order 4
ρ₁₇	2	2	2	2	2	2	-2	-2	-2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	-2	-2	0	0	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	2	2	-2	0	0	0	0	-2	orthogonal lifted from D₄
ρ₁₉	2	2	2	-2	-2	0	0	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	-2	-2	2	0	0	0	0	2	orthogonal lifted from D₄
ρ₂₀	2	2	-2	-2	2	-2	2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	2	2	-2	2	-2	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	-2	2	2	0	orthogonal lifted from D₄
ρ₂₂	2	2	-2	-2	2	2	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₃	2	2	-2	2	-2	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	2	2	-2	-2	0	orthogonal lifted from D₄
ρ₂₄	2	2	2	2	2	-2	2	-2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₅	2	2	2	-2	-2	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	-2i	2i	0	complex lifted from C₄○D₄
ρ₂₆	2	2	-2	2	-2	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	2i	-2i	-2i	0	0	0	0	2i	complex lifted from C₄○D₄
ρ₂₇	2	2	-2	2	-2	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	-2i	2i	2i	0	0	0	0	-2i	complex lifted from C₄○D₄
ρ₂₈	2	2	2	-2	-2	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	2i	-2i	0	complex lifted from C₄○D₄
ρ₂₉	8	-8	0	0	0	0	0	0	0	0	0	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 C₄.(C₄×D₄)
►On 32 points

Generators in S₃₂

(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(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)

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

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

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

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

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

Matrix representation of C₄.(C₄×D₄) ►in GL₈(𝔽₁₇)

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

16	11	7	11	0	0	0	0
16	1	7	10	0	0	0	0
10	6	16	11	0	0	0	0
10	7	16	1	0	0	0	0
7	14	0	3	13	3	4	3
0	14	1	3	14	4	14	13
0	3	10	3	13	14	13	3
1	3	0	3	3	4	14	4

1	0	7	0	8	6	8	11
0	0	0	0	1	16	7	10
10	0	1	0	8	11	9	11
0	0	0	0	7	10	16	1
7	13	0	13	13	3	4	3
7	14	0	3	13	3	4	3
0	13	10	4	13	14	13	3
0	3	10	3	13	14	13	3

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

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

C₄.(C₄×D₄) in GAP, Magma, Sage, TeX

C_4.(C_4\times D_4)

% in TeX

G:=Group("C4.(C4xD4)");

// GroupNames label

G:=SmallGroup(128,641);

// by ID

G=gap.SmallGroup(128,641);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,2804,1411,1027]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄.(C₄×D₄) in TeX

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

16	11	7	11	0	0	0	0
16	1	7	10	0	0	0	0
10	6	16	11	0	0	0	0
10	7	16	1	0	0	0	0
7	14	0	3	13	3	4	3
0	14	1	3	14	4	14	13
0	3	10	3	13	14	13	3
1	3	0	3	3	4	14	4

1	0	7	0	8	6	8	11
0	0	0	0	1	16	7	10
10	0	1	0	8	11	9	11
0	0	0	0	7	10	16	1
7	13	0	13	13	3	4	3
7	14	0	3	13	3	4	3
0	13	10	4	13	14	13	3
0	3	10	3	13	14	13	3

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

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

16	11	7	11	0	0	0	0
16	1	7	10	0	0	0	0
10	6	16	11	0	0	0	0
10	7	16	1	0	0	0	0
7	14	0	3	13	3	4	3
0	14	1	3	14	4	14	13
0	3	10	3	13	14	13	3
1	3	0	3	3	4	14	4

1	0	7	0	8	6	8	11
0	0	0	0	1	16	7	10
10	0	1	0	8	11	9	11
0	0	0	0	7	10	16	1
7	13	0	13	13	3	4	3
7	14	0	3	13	3	4	3
0	13	10	4	13	14	13	3
0	3	10	3	13	14	13	3

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

G = C4.(C4×D4) order 128 = 27

5th non-split extension by C4 of C4×D4 acting via C4×D4/C42=C2

G = C₄.(C₄×D₄) order 128 = 2⁷

5^th non-split extension by C₄ of C₄×D₄ acting via C₄×D₄/C₄²=C₂

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

16	11	7	11	0	0	0	0
16	1	7	10	0	0	0	0
10	6	16	11	0	0	0	0
10	7	16	1	0	0	0	0
7	14	0	3	13	3	4	3
0	14	1	3	14	4	14	13
0	3	10	3	13	14	13	3
1	3	0	3	3	4	14	4

1	0	7	0	8	6	8	11
0	0	0	0	1	16	7	10
10	0	1	0	8	11	9	11
0	0	0	0	7	10	16	1
7	13	0	13	13	3	4	3
7	14	0	3	13	3	4	3
0	13	10	4	13	14	13	3
0	3	10	3	13	14	13	3

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