C2^3:2D4 - GroupNames

G = C₂³:₂D₄ order 64 = 2⁶

1^st semidirect product of C₂³ and D₄ acting via D₄/C₂=C₂²

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂³:₂D₄, C₂⁴.₄C₂², C₂³.₇₅C₂³, (C₂xC₄):₂D₄, C₂.₄C₂²wrC₂, (C₂²xD₄):₁C₂, C₂.₃(C₄:₁D₄), C₂.₆(C₄:D₄), C₂².₆₈(C₂xD₄), C₂.C₄²:₉C₂, (C₂²xC₄).₇C₂², C₂².₃₅(C₄oD₄), (C₂xC₂²:C₄):₆C₂, SmallGroup(64,73)

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

Derived series

C₁ — C₂³ — C₂³:₂D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²xD₄ — C₂³:₂D₄

Lower central

C₁ — C₂³ — C₂³:₂D₄

Upper central

C₁ — C₂³ — C₂³:₂D₄

Jennings

C₁ — C₂³ — C₂³:₂D₄

Generators and relations for C₂³:₂D₄
G = < a,b,c,d,e | a²=b²=c²=d⁴=e²=1, eae=ab=ba, ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 317 in 161 conjugacy classes, 43 normal (7 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, C₂²:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂⁴, C₂.C₄², C₂xC₂²:C₄, C₂²xD₄, C₂³:₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₄oD₄, C₂²wrC₂, C₄:D₄, C₄:₁D₄, C₂³:₂D₄

Character table of C₂³:₂D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	4F	4G	4H
size	1	1	1	1	1	1	1	1	4	4	4	4	4	4	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	trivial
ρ₂	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	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	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	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	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	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	linear of order 2
ρ₉	2	-2	2	2	2	-2	-2	-2	-2	0	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	2	2	2	-2	-2	-2	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	2	0	-2	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	-2	2	-2	2	2	0	0	0	0	0	0	0	2	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	-2	-2	-2	2	-2	0	0	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	-2	-2	-2	2	-2	0	0	2	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	-2	-2	2	-2	2	2	0	0	0	0	0	0	0	-2	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	-2	-2	2	2	-2	-2	0	-2	0	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	2	-2	-2	2	2	-2	-2	0	2	0	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	-2	0	0	2	0	0	orthogonal lifted from D₄
ρ₂₀	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	-2	0	2	0	orthogonal lifted from D₄
ρ₂₁	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	2i	0	0	0	0	0	0	-2i	complex lifted from C₄oD₄
ρ₂₂	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	-2i	0	0	0	0	0	0	2i	complex lifted from C₄oD₄

Smallest permutation representation of C₂³:₂D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

C₂³:₂D₄ is a maximal subgroup of
C₄²:₁₅D₄ C₄²:₁₆D₄ C₂⁴:₇D₄ C₂³.₃₀₄C₂⁴ C₂³.₃₀₈C₂⁴ C₂⁴:₈D₄ C₂⁴.₂₄₉C₂³ C₂³.₃₂₄C₂⁴ C₂³.₃₂₈C₂⁴ C₂⁴.₂₆₂C₂³ C₂³.₃₃₃C₂⁴ C₂⁴.₂₇₆C₂³ C₂³.₃₅₆C₂⁴ C₂⁴.₂₇₈C₂³ C₂³.₃₅₉C₂⁴ C₂⁴.₂₈₂C₂³ C₂⁴.₂₈₃C₂³ C₂³.₃₆₄C₂⁴ C₂³.₃₆₇C₂⁴ C₂⁴.₂₉₀C₂³ C₄²:₁₇D₄ C₂³.₄₃₉C₂⁴ C₄²:₁₉D₄ C₄²:₂₀D₄ C₂³.₄₄₃C₂⁴ C₂⁴.₃₂₇C₂³ C₂³.₄₅₅C₂⁴ C₂³.₄₅₇C₂⁴ C₄²:₂₂D₄ C₂³.₅₀₀C₂⁴ C₂³.₅₀₂C₂⁴ C₄²:₂₄D₄ C₄²:₂₆D₄ C₂⁴:₉D₄ C₂⁴:₁₀D₄ C₄²:₂₇D₄ C₂³.₅₃₀C₂⁴ C₄²:₂₉D₄ C₂³.₅₃₅C₂⁴ C₄²:₃₀D₄ C₂³.₅₄₈C₂⁴ C₂³.₅₅₆C₂⁴ C₄²:₃₁D₄ C₂⁴.₃₇₇C₂³ C₄²:₃₂D₄ C₂³.₅₆₈C₂⁴ C₂³.₅₆₉C₂⁴ C₂³.₅₇₀C₂⁴ C₂³.₅₇₁C₂⁴ C₂³.₅₇₃C₂⁴ C₂⁴.₃₈₄C₂³ C₂³.₅₇₆C₂⁴ C₂³.₅₇₈C₂⁴ C₂⁴.₃₈₉C₂³ C₂⁴.₃₉₅C₂³ C₂³.₅₉₃C₂⁴ C₂⁴.₄₀₃C₂³ C₂⁴.₄₀₆C₂³ C₂³.₆₀₃C₂⁴ C₂³.₆₀₅C₂⁴ C₂³.₆₀₆C₂⁴ C₂⁴.₄₁₂C₂³ C₂³.₆₁₂C₂⁴ C₂⁴.₄₁₃C₂³ C₂³.₆₃₀C₂⁴ C₂³.₆₃₃C₂⁴ C₂³.₆₄₉C₂⁴ C₂³.₆₅₂C₂⁴ C₂⁴.₄₃₇C₂³ C₂³.₆₅₆C₂⁴ C₂³.₆₆₀C₂⁴ C₂³.₆₇₈C₂⁴ C₂³.₆₉₇C₂⁴ C₂³.₇₀₃C₂⁴ C₂⁴:₁₁D₄ C₂⁴.₄₅₉C₂³ C₂³.₇₁₅C₂⁴ C₄²:₃₃D₄ C₄²:₃₄D₄ C₂³.₇₂₄C₂⁴ C₂³.₇₂₅C₂⁴ C₂³.₇₂₆C₂⁴ C₂³.₇₂₈C₂⁴ C₂³.₇₂₉C₂⁴ C₂³:₂D₄:C₃
C₂⁴.D_2p: C₂⁴.D₄ C₂³:D₈ C₂³:SD₁₆ C₂⁴.₉D₄ C₂³:₃D₁₂ C₂⁴.₃₂D₆ C₂³:₂D₂₀ C₂⁴.₂₁D₁₀ ...
C_2p.C₂²wrC₂: C₂³.₂₈₈C₂⁴ C₂⁴.₂₄₄C₂³ C₂⁴.₂₆₃C₂³ C₂⁴.₃₆₀C₂³ (C₂xC₁₂):₅D₄ (C₂xC₂₀):₅D₄ (C₂xC₂₈):₅D₄ ...
C₂³:₂D₄ is a maximal quotient of
C₂⁴.₅₀D₄ C₂⁴.₅Q₈ C₂⁴.₆₃₄C₂³ C₄²:₉D₄ C₄².₁₂₉D₄ C₄²:₁₀D₄ C₄².₁₃₀D₄ M₄(2):D₄ M₄(2):₄D₄ M₄(2).D₄ (C₂xC₄):₃SD₁₆ (C₂xC₄):₂Q₁₆ (C₂xC₈).₂D₄ M₄(2).₄D₄ C₂⁴:D₄ C₂⁴.₃₁D₄
(C₂xC_4p):D₄: C₂³:₂D₈ C₂³:₃SD₁₆ M₄(2):₅D₄ C₄²:₂D₄ (C₂xC₄):₂D₈ (C₂xC₈):₂₀D₄ (C₂xC₁₂):₅D₄ C₂³:₃D₁₂ ...
C₂.(D₄._pD₄): C₂³:₂Q₁₆ (C₂²xD₈).C₂ (C₂xC₈).₄₁D₄ M₄(2).₅D₄ M₄(2).₆D₄ ...

Matrix representation of C₂³:₂D₄ ►in GL₆(F₅)

4	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	2
0	0	0	0	0	1

4	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

C₂³:₂D₄ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_4

% in TeX

G:=Group("C2^3:2D4");

// GroupNames label

G:=SmallGroup(64,73);

// by ID

G=gap.SmallGroup(64,73);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,121,362,332]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³:₂D₄ in TeX

G = C23:2D4 order 64 = 26

1st semidirect product of C23 and D4 acting via D4/C2=C22

G = C₂³:₂D₄ order 64 = 2⁶

1^st semidirect product of C₂³ and D₄ acting via D₄/C₂=C₂²