(C2xC4):D8 - GroupNames

G = (C₂×C₄)⋊D₈ order 128 = 2⁷

The semidirect product of C₂×C₄ and D₈ acting via D₈/C₂=D₄

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

Aliases: (C₂×C₄)⋊D₈, C₄⋊C₄.₂D₄, C₂²⋊D₈⋊₂C₂, C₂.₇C₂≀C₂², (C₂×D₄).₁₀D₄, (C₂²×C₄).₄₂D₄, C₂².₁₄(C₂×D₈), C₂.₇(C₂²⋊D₈), C₂³.₅₁₅(C₂×D₄), C₂².SD₁₆⋊₅C₂, C₄⋊D₄.₃C₂², C₂²⋊C₈.₁C₂², (C₂²×C₄).₄C₂³, C₂.₇(D₄.₈D₄), C₂³.₁₀D₄⋊₂C₂, (C₂²×D₄).₅C₂², C₂².₁₂₅C₂²≀C₂, C₂².₃₉(C₈⋊C₂²), C₂².₃₁C₂⁴⋊₁C₂, C₂².M₄(2)⋊₃C₂, C₂.C₄².₁₃C₂², (C₂×C₄).₁₉₃(C₂×D₄), (C₂×C₄⋊C₄).₁₄C₂², SmallGroup(128,330)

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

Derived series

C₁ — C₂²×C₄ — (C₂×C₄)⋊D₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂².₃₁C₂⁴ — (C₂×C₄)⋊D₈

Lower central

C₁ — C₂² — C₂²×C₄ — (C₂×C₄)⋊D₈

Upper central

C₁ — C₂² — C₂²×C₄ — (C₂×C₄)⋊D₈

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — (C₂×C₄)⋊D₈

Generators and relations for (C₂×C₄)⋊D₈
G = < a,b,c,d | a²=b⁴=c⁸=d²=1, cbc^-1=dbd=ab=ba, cac^-1=ab², ad=da, dcd=c^-1 >

Subgroups: 452 in 155 conjugacy classes, 34 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, D₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂.C₄², C₂²⋊C₈, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂×D₈, C₂²×D₄, C₂×C₄○D₄, C₂².M₄(2), C₂².SD₁₆, C₂³.₁₀D₄, C₂²⋊D₈, C₂².₃₁C₂⁴, (C₂×C₄)⋊D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₂²≀C₂, C₂×D₈, C₈⋊C₂², C₂²⋊D₈, D₄.₈D₄, C₂≀C₂², (C₂×C₄)⋊D₈

Character table of (C₂×C₄)⋊D₈

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D
size	1	1	1	1	2	2	8	8	8	8	4	4	4	4	8	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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	0	2	0	0	2	-2	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	0	0	0	0	2	-2	-2	2	0	-2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	0	-2	0	0	2	-2	0	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	-2	0	0	0	-2	2	0	0	0	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	2	0	0	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	0	2	0	0	0	-2	2	0	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	-2	2	0	0	0	0	2	0	0	-2	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₆	2	2	-2	-2	-2	2	0	0	0	0	-2	0	0	2	0	0	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	-2	2	0	0	0	0	2	0	0	-2	0	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₈	2	2	-2	-2	-2	2	0	0	0	0	-2	0	0	2	0	0	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₉	4	-4	4	-4	0	0	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂≀C₂²
ρ₂₀	4	-4	4	-4	0	0	-2	0	0	2	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂≀C₂²
ρ₂₁	4	4	-4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	0	0	0	0	complex lifted from D₄.₈D₄
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	-2i	0	0	0	2i	0	0	0	0	complex lifted from D₄.₈D₄

Smallest permutation representation of (C₂×C₄)⋊D₈
►On 32 points

Generators in S₃₂

(1 32)(3 26)(5 28)(7 30)(10 17)(12 19)(14 21)(16 23)

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

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

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

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

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

Matrix representation of (C₂×C₄)⋊D₈ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	0	6	14
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	15	3	14	8
0	0	4	2	5	6
0	0	0	0	13	0
0	0	0	0	15	4

8	11	0	0	0	0
0	15	0	0	0	0
0	0	1	16	10	2
0	0	0	0	8	1
0	0	8	12	10	12
0	0	4	2	5	6

16	0	0	0	0	0
4	1	0	0	0	0
0	0	15	12	8	4
0	0	4	2	5	6
0	0	0	0	16	0
0	0	0	0	8	1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,6,0,1,0,0,0,14,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,4,0,0,0,0,3,2,0,0,0,0,14,5,13,15,0,0,8,6,0,4],[8,0,0,0,0,0,11,15,0,0,0,0,0,0,1,0,8,4,0,0,16,0,12,2,0,0,10,8,10,5,0,0,2,1,12,6],[16,4,0,0,0,0,0,1,0,0,0,0,0,0,15,4,0,0,0,0,12,2,0,0,0,0,8,5,16,8,0,0,4,6,0,1] >;

(C₂×C₄)⋊D₈ in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes D_8

% in TeX

G:=Group("(C2xC4):D8");

// GroupNames label

G:=SmallGroup(128,330);

// by ID

G=gap.SmallGroup(128,330);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,352,1123,570,521,136,1411]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,d*c*d=c^-1>;

// generators/relations

Export

Character table of (C₂×C₄)⋊D₈ in TeX

G = (C2×C4)⋊D8 order 128 = 27

The semidirect product of C2×C4 and D8 acting via D8/C2=D4

G = (C₂×C₄)⋊D₈ order 128 = 2⁷

The semidirect product of C₂×C₄ and D₈ acting via D₈/C₂=D₄