C2^3.11D4 - GroupNames

G = C₂³.₁₁D₄ order 64 = 2⁶

4^th non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂³.₁₁D₄, C₂⁴.₈C₂², C₂³.₈₀C₂³, (C₂×C₄).₁₇D₄, C₂.₉(C₄⋊D₄), C₂².₇₃(C₂×D₄), C₂.C₄²⋊₅C₂, C₂.₇(C₄.₄D₄), (C₂²×C₄).₉C₂², C₂.₅(C₄²⋊₂C₂), C₂².₄₀(C₄○D₄), C₂.₇(C₂².D₄), (C₂×C₄⋊C₄)⋊₇C₂, (C₂×C₂²⋊C₄).₉C₂, SmallGroup(64,78)

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

Derived series

C₁ — C₂³ — C₂³.₁₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³.₁₁D₄

Lower central

C₁ — C₂³ — C₂³.₁₁D₄

Upper central

C₁ — C₂³ — C₂³.₁₁D₄

Jennings

C₁ — C₂³ — C₂³.₁₁D₄

Generators and relations for C₂³.₁₁D₄
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e²=b, dad^-1=ab=ba, ac=ca, eae^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 157 in 85 conjugacy classes, 35 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³.₁₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂³.₁₁D₄

Character table of C₂³.₁₁D₄

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

Smallest permutation representation of C₂³.₁₁D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

Matrix representation of C₂³.₁₁D₄ ►in GL₆(𝔽₅)

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

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

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

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

C₂³.₁₁D₄ in GAP, Magma, Sage, TeX

C_2^3._{11}D_4

% in TeX

G:=Group("C2^3.11D4");

// GroupNames label

G:=SmallGroup(64,78);

// by ID

G=gap.SmallGroup(64,78);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,362,332,50]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.₁₁D₄ in TeX

G = C23.11D4 order 64 = 26

4th non-split extension by C23 of D4 acting via D4/C2=C22

G = C₂³.₁₁D₄ order 64 = 2⁶

4^th non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²