C2^3.35D8 - GroupNames

G = C₂³.₃₅D₈ order 128 = 2⁷

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

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

Aliases: C₂³.₃₅D₈, C₂⁴.₁₅₄D₄, C₂³.₃₂SD₁₆, D₄⋊₃(C₂²⋊C₄), C₄.₅₂C₂²≀C₂, (C₂×D₄).₂₆₃D₄, (C₂²×D₄)⋊₁₇C₄, (C₂²×C₈)⋊₂C₂², (D₄×C₂³).₄C₂, C₂².₃₀(C₂×D₈), C₂.₁(C₂²⋊D₈), C₂³.₇₃₇(C₂×D₄), (C₂²×C₄).₂₆₂D₄, C₂³.₇Q₈⋊₃C₂, C₂²⋊₃(D₄⋊C₄), C₂.₁(C₂²⋊SD₁₆), C₂².₇₂C₂²≀C₂, C₂².₄₄(C₂×SD₁₆), C₂.₁₁(C₂⁴⋊₃C₄), C₂².₅₇(C₈⋊C₂²), (C₂³×C₄).₂₃₁C₂², C₂³.₁₉₄(C₂²⋊C₄), (C₂²×C₄).₁₃₂₀C₂³, (C₂²×D₄).₄₅₂C₂², C₂.₁₇(C₂³.₃₇D₄), (C₂×C₄⋊C₄)⋊₃C₂², C₄.₁(C₂×C₂²⋊C₄), (C₂×D₄⋊C₄)⋊₁C₂, (C₂×C₂²⋊C₈)⋊₁₁C₂, (C₂×D₄).₂₀₀(C₂×C₄), C₂.₁₇(C₂×D₄⋊C₄), (C₂×C₄).₁₃₁₀(C₂×D₄), (C₂²×C₄).₂₆₀(C₂×C₄), (C₂×C₄).₃₅₈(C₂²×C₄), (C₂×C₄).₁₁₉(C₂²⋊C₄), C₂².₂₃₉(C₂×C₂²⋊C₄), SmallGroup(128,518)

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

Derived series

C₁ — C₂×C₄ — C₂³.₃₅D₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — D₄×C₂³ — C₂³.₃₅D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂³.₃₅D₈

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂³.₃₅D₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³.₃₅D₈

Generators and relations for C₂³.₃₅D₈
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=c, dad^-1=eae^-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 1052 in 424 conjugacy classes, 84 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂²⋊C₈, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, D₄×C₂³, C₂³.₃₅D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, D₈, SD₁₆, C₂²×C₄, C₂×D₄, D₄⋊C₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₂⁴⋊₃C₄, C₂×D₄⋊C₄, C₂³.₃₇D₄, C₂²⋊D₈, C₂²⋊SD₁₆, C₂³.₃₅D₈

Smallest permutation representation of C₂³.₃₅D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2S	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	···	8H
order	1	2	···	2	2	2	2	2	2	···	2	4	4	4	4	4	4	4	4	4	4	8	···	8
size	1	1	···	1	2	2	2	2	4	···	4	2	2	2	2	4	4	8	8	8	8	4	···	4

38 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₈

SD₁₆

C₈⋊C₂²

kernel

C₂³.₃₅D₈

C₂³.₇Q₈

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

D₄×C₂³

C₂²×D₄

C₂²×C₄

C₂×D₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂³.₃₅D₈ ►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	5	16

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	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	1	0
0	0	0	0	0	1

3	14	0	0	0	0
3	3	0	0	0	0
0	0	0	2	0	0
0	0	9	0	0	0
0	0	0	0	11	16
0	0	0	0	1	6

3	3	0	0	0	0
3	14	0	0	0	0
0	0	0	2	0	0
0	0	8	0	0	0
0	0	0	0	6	1
0	0	0	0	16	11

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,5,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,9,0,0,0,0,2,0,0,0,0,0,0,0,11,1,0,0,0,0,16,6],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,8,0,0,0,0,2,0,0,0,0,0,0,0,6,16,0,0,0,0,1,11] >;

C₂³.₃₅D₈ in GAP, Magma, Sage, TeX

C_2^3._{35}D_8

% in TeX

G:=Group("C2^3.35D8");

// GroupNames label

G:=SmallGroup(128,518);

// by ID

G=gap.SmallGroup(128,518);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,248]);

// 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,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;

// generators/relations

G = C23.35D8 order 128 = 27

6th non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₃₅D₈ order 128 = 2⁷

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