C2^3.10D4 - GroupNames

G = C₂³.₁₀D₄ order 64 = 2⁶

3^rd non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂³.₁₀D₄, C₂⁴.₆C₂², C₂³.₇₇C₂³, (C₂×C₄)⋊₃D₄, C₂.₆C₂²≀C₂, C₂.₇(C₄⋊D₄), (C₂²×D₄).₃C₂, C₂².₇₀(C₂×D₄), C₂.₆(C₄.₄D₄), C₂.C₄²⋊₁₁C₂, C₂².₃₇(C₄○D₄), (C₂²×C₄).₂₆C₂², C₂.₆(C₂².D₄), (C₂×C₄⋊C₄)⋊₅C₂, (C₂×C₂²⋊C₄)⋊₇C₂, SmallGroup(64,75)

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

Derived series

C₁ — C₂³ — C₂³.₁₀D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³.₁₀D₄

Lower central

C₁ — C₂³ — C₂³.₁₀D₄

Upper central

C₁ — C₂³ — C₂³.₁₀D₄

Jennings

C₁ — C₂³ — C₂³.₁₀D₄

Generators and relations for C₂³.₁₀D₄
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=bd^-1 >

Subgroups: 229 in 119 conjugacy classes, 39 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×D₄, C₂³.₁₀D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂³.₁₀D₄

Character table of C₂³.₁₀D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J
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	0	2	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	0	0	0	0	0	0	2	0	0	0	0	0	0	-2	orthogonal lifted from D₄
ρ₁₁	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	2	0	0	0	0	0	0	-2	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	-2	-2	-2	2	-2	2	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	-2	0	0	0	0	0	0	2	0	orthogonal lifted from D₄
ρ₁₄	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	-2	0	0	0	0	0	0	2	orthogonal lifted from D₄
ρ₁₅	2	2	2	-2	-2	-2	2	-2	-2	0	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	-2	-2	2	2	-2	-2	0	-2	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	-2	-2	-2	2	-2	2	2	0	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	-2	2	-2	2	2	0	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	2	2	-2	-2	-2	0	0	0	0	2i	0	0	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	2	2	2	-2	-2	-2	0	0	0	0	-2i	0	0	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	0	complex lifted from C₄○D₄

Smallest permutation representation of C₂³.₁₀D₄
►On 32 points

Generators in S₃₂

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

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

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

(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 21)(4 23)(5 16)(6 8)(7 14)(9 11)(10 32)(12 30)(13 15)(17 25)(19 27)(29 31)

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

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

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

Matrix representation of C₂³.₁₀D₄ ►in GL₆(𝔽₅)

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

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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	4	0	0
0	0	4	0	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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

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

C₂³.₁₀D₄ in GAP, Magma, Sage, TeX

C_2^3._{10}D_4

% in TeX

G:=Group("C2^3.10D4");

// GroupNames label

G:=SmallGroup(64,75);

// by ID

G=gap.SmallGroup(64,75);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,121,362,332,50]);

// 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=b*d^-1>;

// generators/relations

Export

Character table of C₂³.₁₀D₄ in TeX

G = C23.10D4 order 64 = 26

3rd non-split extension by C23 of D4 acting via D4/C2=C22

G = C₂³.₁₀D₄ order 64 = 2⁶

3^rd non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²