(C2xQ8).D4 - GroupNames

G = (C₂×Q₈).D₄ order 128 = 2⁷

6^th non-split extension by C₂×Q₈ of D₄ acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: (C₄×C₈).₄C₄, C₄⋊Q₈.₄C₄, (C₂×Q₈).₆D₄, C₄⋊Q₈.₃C₂², C₄².₁₈(C₂×C₄), C₄⋊Q₁₆.₂C₂, C₄².₃C₄.₁C₂, C₂.₁₁(C₄²⋊C₄), C₂².₂₁(C₂³⋊C₄), (C₂×C₄).₃₇(C₂²⋊C₄), SmallGroup(128,143)

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

Derived series

C₁ — C₄² — (C₂×Q₈).D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂×Q₈ — C₄⋊Q₈ — C₄⋊Q₁₆ — (C₂×Q₈).D₄

Lower central

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — (C₂×Q₈).D₄

Upper central

C₁ — C₂ — C₂² — C₂×C₄ — C₄⋊Q₈ — (C₂×Q₈).D₄

Jennings

C₁ — C₂ — C₂ — C₂² — C₂×C₄ — C₄⋊Q₈ — (C₂×Q₈).D₄

Generators and relations for (C₂×Q₈).D₄
G = < a,b,c,d,e | a²=b⁴=1, c²=d⁴=b², e²=ca=ac, dbd^-1=ab=ba, dad^-1=eae^-1=ab², cbc^-1=b^-1, ebe^-1=ab^-1, dcd^-1=bc, ece^-1=b²c, ede^-1=acd³ >

C₁

C₂

₂C₂

C₂²

₂C₄

₄C₄

₈C₄

C₂×C₄

₂C₈

₂C₂×C₄

₂C₈

₂C₂×C₄

₄Q₈

₄C₂×C₄

₄Q₈

₈C₈

C₂×Q₈

C₄²

C₂×Q₈

₂C₂×Q₈

₂C₂×C₈

₄Q₁₆

₄M₄(2)

₄C₄⋊C₄

₄Q₁₆

₄C₄⋊C₄

₄M₄(2)

C₄⋊Q₈

C₄×C₈

C₄⋊Q₈

₂C₂×Q₁₆

₂C₄.₁₀D₄

₂C₂×Q₁₆

₂C₄.₁₀D₄

C₄².₃C₄

C₄⋊Q₁₆

(C₂×Q₈).D₄

Character table of (C₂×Q₈).D₄

class	1	2A	2B	4A	4B	4C	4D	4E	4F	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	2	4	4	4	8	8	16	4	4	4	4	16	16	16	16
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	1	1	1	1	1	1	-1	-1	-1	1	1	1	1	i	i	-i	-i	linear of order 4
ρ₆	1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	-i	i	i	-i	linear of order 4
ρ₇	1	1	1	1	1	1	-1	-1	-1	1	1	1	1	-i	-i	i	i	linear of order 4
ρ₈	1	1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	i	-i	-i	i	linear of order 4
ρ₉	2	2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	4	4	-4	0	0	0	0	0	0	2	2	-2	-2	0	0	0	0	orthogonal lifted from C₄²⋊C₄
ρ₁₂	4	4	-4	0	0	0	0	0	0	-2	-2	2	2	0	0	0	0	orthogonal lifted from C₄²⋊C₄
ρ₁₃	4	4	4	0	-4	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₁₄	4	-4	0	2	0	-2	0	0	0	-2√2	2√2	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₁₅	4	-4	0	-2	0	2	0	0	0	0	0	2√2	-2√2	0	0	0	0	symplectic faithful, Schur index 2
ρ₁₆	4	-4	0	2	0	-2	0	0	0	2√2	-2√2	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₁₇	4	-4	0	-2	0	2	0	0	0	0	0	-2√2	2√2	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of (C₂×Q₈).D₄
►On 32 points

Generators in S₃₂

(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)

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

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

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

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

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

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

Matrix representation of (C₂×Q₈).D₄ ►in GL₄(𝔽₇) generated by

0	6	3	2
6	0	4	2
0	0	6	0
0	0	0	1

2	4	1	2
3	2	3	4
1	1	5	5
5	2	1	5

2	3	0	3
0	1	5	5
4	4	4	6
3	4	2	0

6	0	6	2
2	3	0	1
0	0	0	4
1	1	1	5

1	0	5	6
4	0	2	5
4	3	5	3
3	3	5	1

G:=sub<GL(4,GF(7))| [0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[2,3,1,5,4,2,1,2,1,3,5,1,2,4,5,5],[2,0,4,3,3,1,4,4,0,5,4,2,3,5,6,0],[6,2,0,1,0,3,0,1,6,0,0,1,2,1,4,5],[1,4,4,3,0,0,3,3,5,2,5,5,6,5,3,1] >;

(C₂×Q₈).D₄ in GAP, Magma, Sage, TeX

(C_2\times Q_8).D_4

% in TeX

G:=Group("(C2xQ8).D4");

// GroupNames label

G:=SmallGroup(128,143);

// by ID

G=gap.SmallGroup(128,143);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,1059,520,794,745,248,1684,1411,375,172,4037]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^4=b^2,e^2=c*a=a*c,d*b*d^-1=a*b=b*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c^-1=b^-1,e*b*e^-1=a*b^-1,d*c*d^-1=b*c,e*c*e^-1=b^2*c,e*d*e^-1=a*c*d^3>;

// generators/relations

Export

Subgroup lattice of (C₂×Q₈).D₄ in TeX
Character table of (C₂×Q₈).D₄ in TeX

G = (C2×Q8).D4 order 128 = 27

6th non-split extension by C2×Q8 of D4 acting faithfully

G = (C₂×Q₈).D₄ order 128 = 2⁷

6^th non-split extension by C₂×Q₈ of D₄ acting faithfully