C2^3.5D8 - GroupNames

G = C₂³.₅D₈ order 128 = 2⁷

5^th non-split extension by C₂³ of D₈ acting via D₈/C₂=D₄

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

Aliases: C₂³.₅D₈, C₂⁴.₁₃D₄, C₂³⋊C₈⋊₄C₂, C₄⋊C₄.₁₀D₄, (C₂×D₄).₁₅D₄, C₂²⋊C₈⋊₂C₂², (C₂²×C₄).₄₈D₄, C₂².₁₅(C₂×D₈), C₂.₈(C₂²⋊D₈), C₂³⋊₃D₄.₂C₂, C₂².D₈⋊₁C₂, C₂³.₅₂₄(C₂×D₄), C₂².SD₁₆⋊₇C₂, C₄⋊D₄.₉C₂², C₂.₉(D₄.₉D₄), C₂³.₁₁D₄⋊₁C₂, (C₂²×C₄).₁₃C₂³, C₂².₁₃₄C₂²≀C₂, C₂².₄₂(C₈⋊C₂²), C₂.C₄²⋊₅C₂², C₂.₄(C₂³.₇D₄), (C₂×C₄⋊C₄)⋊₁C₂², (C₂×C₄).₂₀₂(C₂×D₄), (C₂×C₂²⋊C₄).₉₆C₂², SmallGroup(128,339)

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

Derived series

C₁ — C₂²×C₄ — C₂³.₅D₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³⋊₃D₄ — C₂³.₅D₈

Lower central

C₁ — C₂² — C₂²×C₄ — C₂³.₅D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂³.₅D₈

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂³.₅D₈

Generators and relations for C₂³.₅D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=c, dad^-1=eae^-1=ab=ba, ac=ca, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 412 in 140 conjugacy classes, 34 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, D₄⋊C₄, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₂³⋊C₈, C₂².SD₁₆, C₂³.₁₁D₄, C₂².D₈, C₂³⋊₃D₄, C₂³.₅D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₂²≀C₂, C₂×D₈, C₈⋊C₂², C₂²⋊D₈, D₄.₉D₄, C₂³.₇D₄, C₂³.₅D₈

Character table of C₂³.₅D₈

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	4	4	8	8	4	4	8	8	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	0	2	2	-2	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	0	0	-2	0	-2	2	0	0	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	2	-2	-2	0	0	-2	-2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	0	0	-2	2	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	0	0	2	0	-2	2	0	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	2	2	2	0	0	-2	-2	0	-2	0	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	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₆	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₈	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₉	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
ρ₂₀	4	-4	-4	4	0	0	0	0	0	0	0	0	-2i	0	0	0	0	0	2i	0	0	0	0	complex lifted from D₄.₉D₄
ρ₂₁	4	-4	-4	4	0	0	0	0	0	0	0	0	2i	0	0	0	0	0	-2i	0	0	0	0	complex lifted from D₄.₉D₄
ρ₂₂	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	0	0	0	complex lifted from C₂³.₇D₄

Smallest permutation representation of C₂³.₅D₈
►On 32 points

Generators in S₃₂

(2 16)(3 20)(4 30)(6 12)(7 24)(8 26)(9 29)(10 21)(13 25)(14 17)(19 28)(23 32)

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

(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(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)

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

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

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

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

Matrix representation of C₂³.₅D₈ ►in GL₆(𝔽₁₇)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	15	0	0
0	0	0	16	0	0
0	0	0	1	0	1
0	0	0	1	1	0

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	0	0	16	0
0	0	0	0	0	16

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

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

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

C₂³.₅D₈ in GAP, Magma, Sage, TeX

C_2^3._5D_8

% in TeX

G:=Group("C2^3.5D8");

// GroupNames label

G:=SmallGroup(128,339);

// by ID

G=gap.SmallGroup(128,339);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,1123,570,521,136,1411]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.₅D₈ in TeX

G = C23.5D8 order 128 = 27

5th non-split extension by C23 of D8 acting via D8/C2=D4

G = C₂³.₅D₈ order 128 = 2⁷

5^th non-split extension by C₂³ of D₈ acting via D₈/C₂=D₄