C2^2.D8 - GroupNames

G = C₂².D₈ order 64 = 2⁶

3^rd non-split extension by C₂² of D₈ acting via D₈/D₄=C₂

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

Aliases: C₂².₄D₈, C₂³.₄₅D₄, C₂.D₈⋊₆C₂, C₂.₈(C₂×D₈), C₂²⋊C₈⋊₄C₂, (C₂×C₄).₃₇D₄, D₄⋊C₄⋊₇C₂, C₄⋊D₄.₅C₂, (C₂×C₈).₇C₂², C₄.₂₉(C₄○D₄), C₄⋊C₄.₆₃C₂², (C₂×C₄).₁₀₅C₂³, (C₂×D₄).₂₁C₂², C₂².₁₀₁(C₂×D₄), C₂.₁₇(C₈.C₂²), (C₂²×C₄).₅₁C₂², C₂.₁₁(C₂².D₄), (C₂×C₄⋊C₄)⋊₁₁C₂, SmallGroup(64,161)

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

Derived series

C₁ — C₂×C₄ — C₂².D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₄⋊C₄ — C₂×C₄⋊C₄ — C₂².D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂².D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂².D₈

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂².D₈

Generators and relations for C₂².D₈
G = < a,b,c,d | a²=b²=c⁸=d²=1, cac^-1=dad=ab=ba, bc=cb, bd=db, dcd=bc^-1 >

Subgroups: 113 in 57 conjugacy classes, 27 normal (15 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂²⋊C₈, D₄⋊C₄, C₂.D₈, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₂².D₄, C₂×D₈, C₈.C₂², C₂².D₈

Character table of C₂².D₈

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D
size	1	1	1	1	2	2	8	2	2	4	4	4	4	4	8	4	4	4	4
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	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	linear of order 2
ρ₇	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	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	-2	-2	0	0	0	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	0	-2	-2	0	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₃	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₄	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	2	-2	0	0	0	-2	2	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₆	2	-2	2	-2	0	0	0	2	-2	0	0	-2i	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₇	2	-2	2	-2	0	0	0	2	-2	0	0	2i	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	0	0	0	-2	2	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of C₂².D₈
►On 32 points

Generators in S₃₂

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

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

(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)

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

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

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

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

C₂².D₈ is a maximal subgroup of
C₂⁴.₁₁₅D₄ C₂⁴.₁₈₃D₄ C₂⁴.₁₁₇D₄ (C₂×D₄).₃₀₁D₄ (C₂×D₄).₃₀₃D₄ C₄².₂₂₁D₄ C₄².₂₂₆D₄ C₄².₂₃₀D₄ C₄².₂₃₂D₄ C₂³⋊₃D₈ C₂⁴.₁₂₃D₄ C₂⁴.₁₂₆D₄ C₂⁴.₁₂₇D₄ C₄.2₊¹⁺⁴ C₄.₁₈2₊¹⁺⁴ C₄².₂₈₄D₄ C₄².₂₈₆D₄ C₄².₂₉₀D₄
(C₂×C_2p).D₈: (C₂×C₄).₅D₈ C₄².₂₇₈D₄ C₂².D₂₄ (C₂×C₆).₄₀D₈ (C₂×C₆).D₈ C₂².D₄₀ (C₂×C₁₀).₄₀D₈ (C₂×C₁₀).D₈ ...
C₄⋊C₄.D_2p: C₂³.₅D₈ C₄⋊C₄.₁₂D₄ C₂⁴.₁₅D₄ C₄².₃₅₃C₂³ C₄².₃₅₈C₂³ C₄².₄₂₃C₂³ C₄².₄₂₅C₂³ D₄⋊₄D₈ ...
C₂².D₈ is a maximal quotient of
C₂³.₃₆D₈ C₂³.₃₇D₈ C₂³.₃₈D₈ C₂⁴.₈₃D₄
(C₂×C_2p).D₈: C₂.D₈⋊₄C₄ D₄⋊C₄⋊C₄ (C₂×C₄).₂₄D₈ (C₂×C₈).₁Q₈ (C₂×C₄).₂₇D₈ (C₂×C₄).₂₈D₈ C₂².D₁₆ C₂³.₄₉D₈ ...
(C₂×C₈).D_2p: C₂³.₁₂D₈ D₆.D₈ D₆.₅D₈ D₁₀.₁₂D₈ D₁₀.₁₃D₈ D₁₄.D₈ D₁₄.₅D₈ ...

Matrix representation of C₂².D₈ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	0	1
0	0	1	0

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

3	14	0	0
3	3	0	0
0	0	4	0
0	0	0	13

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

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[3,3,0,0,14,3,0,0,0,0,4,0,0,0,0,13],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C₂².D₈ in GAP, Magma, Sage, TeX

C_2^2.D_8

% in TeX

G:=Group("C2^2.D8");

// GroupNames label

G:=SmallGroup(64,161);

// by ID

G=gap.SmallGroup(64,161);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,50,1444,376,88]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂².D₈ in TeX

G = C22.D8 order 64 = 26

3rd non-split extension by C22 of D8 acting via D8/D4=C2

G = C₂².D₈ order 64 = 2⁶

3^rd non-split extension by C₂² of D₈ acting via D₈/D₄=C₂