C2^3.40D8 - GroupNames

G = C₂³.₄₀D₈ order 128 = 2⁷

11^st non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂³.₄₀D₈, (C₂×D₈)⋊₁₄C₄, D₈.₈(C₂×C₄), C₈.₉₄(C₂×D₄), (C₂×C₈).₁₁₈D₄, (C₂×C₄).₁₃₈D₈, C₂.D₁₆⋊₁₅C₂, (C₂×C₁₆)⋊₁₁C₂², C₈.₅(C₂²⋊C₄), C₈.₃₁(C₂²×C₄), (C₂×C₄).₄₈SD₁₆, C₄.₁₂(C₂×SD₁₆), C₂².₅₅(C₂×D₈), C₂.D₈⋊₄₃C₂², C₂.₂(C₁₆⋊C₂²), (C₂×M₅(2))⋊₁₃C₂, (C₂×C₈).₄₉₆C₂³, (C₂²×D₈).₁₃C₂, (C₂²×C₄).₃₃₁D₄, C₄.₁₁(D₄⋊C₄), (C₂×D₈).₁₀₂C₂², C₂³.₂₅D₄⋊₁₈C₂, (C₂²×C₈).₂₃₀C₂², C₂².₂₉(D₄⋊C₄), (C₂×C₈).₈₀(C₂×C₄), (C₂×C₄).₇₅₈(C₂×D₄), C₄.₅₂(C₂×C₂²⋊C₄), C₂.₃₀(C₂×D₄⋊C₄), (C₂×C₄).₁₄₉(C₂²⋊C₄), SmallGroup(128,872)

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

Derived series

C₁ — C₈ — C₂³.₄₀D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂²×D₈ — C₂³.₄₀D₈

Lower central

C₁ — C₂ — C₄ — C₈ — C₂³.₄₀D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₄₀D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₂³.₄₀D₈

Generators and relations for C₂³.₄₀D₈
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=c, e²=b, ab=ba, dad^-1=eae^-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bcd⁷ >

Subgroups: 388 in 130 conjugacy classes, 52 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, D₈, C₂²×C₄, C₂×D₄, C₂⁴, C₄.Q₈, C₂.D₈, C₂×C₁₆, M₅(2), C₄²⋊C₂, C₂²×C₈, C₂×D₈, C₂×D₈, C₂²×D₄, C₂.D₁₆, C₂³.₂₅D₄, C₂×M₅(2), C₂²×D₈, C₂³.₄₀D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, D₈, SD₁₆, C₂²×C₄, C₂×D₄, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×D₈, C₂×SD₁₆, C₂×D₄⋊C₄, C₁₆⋊C₂², C₂³.₄₀D₈

Smallest permutation representation of C₂³.₄₀D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	16A	···	16H
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	8	8	8	8	8	8	16	···	16
size	1	1	1	1	2	2	8	8	8	8	2	2	2	2	8	8	8	8	2	2	2	2	4	4	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₈

SD₁₆

D₈

C₁₆⋊C₂²

kernel

C₂³.₄₀D₈

C₂.D₁₆

C₂³.₂₅D₄

C₂×M₅(2)

C₂²×D₈

C₂×D₈

C₂×C₈

C₂²×C₄

C₂×C₄

C₂³

C₂

# reps

Matrix representation of C₂³.₄₀D₈ ►in GL₆(𝔽₁₇)

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	1	0
0	0	0	0	0	1

16	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	0
0	0	0	0	0	16

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

10	9	0	0	0	0
6	7	0	0	0	0
0	0	0	0	0	16
0	0	0	0	16	0
0	0	3	14	0	0
0	0	14	14	0	0

7	8	0	0	0	0
15	10	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	0	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[10,6,0,0,0,0,9,7,0,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,16,0,0,0,0,16,0,0,0],[7,15,0,0,0,0,8,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³.₄₀D₈ in GAP, Magma, Sage, TeX

C_2^3._{40}D_8

% in TeX

G:=Group("C2^3.40D8");

// GroupNames label

G:=SmallGroup(128,872);

// by ID

G=gap.SmallGroup(128,872);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,352,1123,570,360,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;

// generators/relations

G = C23.40D8 order 128 = 27

11st non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₄₀D₈ order 128 = 2⁷

11^st non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂