C2^4.178D4 - GroupNames

G = C₂⁴.₁₇₈D₄ order 128 = 2⁷

33^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂²=C₂

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

Aliases: C₂⁴.₁₇₈D₄, (C₂×Q₈)⋊₃₆D₄, Q₈⋊D₄⋊₁C₂, C₄⋊C₄.₆C₂³, (C₂×C₈).₇C₂³, Q₈.₄₀(C₂×D₄), (Q₈×C₂³)⋊₆C₂, C₄.₈₂C₂²≀C₂, C₄.₄₁(C₂²×D₄), (C₂×C₄).₂₂₃C₂⁴, C₂⁴.₄C₄⋊₃C₂, C₂²⋊Q₁₆⋊₁₁C₂, (C₂×Q₁₆)⋊₁₂C₂², (C₂×SD₁₆)⋊₂C₂², (C₂×D₄).₂₆C₂³, (C₂²×C₄).₄₂₀D₄, C₂³.₆₄₇(C₂×D₄), (C₂×Q₈).₁₉C₂³, Q₈⋊C₄⋊₁₁C₂², C₂².₅₈C₂²≀C₂, C₂³.₃₈D₄⋊₂C₂, C₂²⋊C₈.₁₁C₂², C₂²⋊₄(C₈.C₂²), C₄⋊D₄.₁₄₅C₂², (C₂²×C₄).₉₆₁C₂³, (C₂³×C₄).₅₄₃C₂², C₂².₄₈₃(C₂²×D₄), C₂²⋊Q₈.₁₅₀C₂², C₂².₁₉C₂⁴.₁₅C₂, C₄²⋊C₂.₉₅C₂², (C₂×M₄(2)).₃₉C₂², (C₂²×Q₈).₄₆₇C₂², (C₂×C₄).₄₅₁(C₂×D₄), (C₂×C₈.C₂²)⋊₅C₂, C₂.₄₁(C₂×C₂²≀C₂), C₂.₁₀(C₂×C₈.C₂²), (C₂×C₄○D₄).₉₉C₂², SmallGroup(128,1736)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₇₈D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — Q₈×C₂³ — C₂⁴.₁₇₈D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₇₈D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₇₈D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₇₈D₄

Generators and relations for C₂⁴.₁₇₈D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=f²=1, e⁴=d, ab=ba, eae^-1=faf=ac=ca, ad=da, bc=cb, ebe^-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e³ >

Subgroups: 660 in 369 conjugacy classes, 112 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊C₈, Q₈⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂⁴.₄C₄, C₂³.₃₈D₄, Q₈⋊D₄, C₂²⋊Q₁₆, C₂².₁₉C₂⁴, C₂×C₈.C₂², Q₈×C₂³, C₂⁴.₁₇₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₈.C₂², C₂²×D₄, C₂×C₂²≀C₂, C₂×C₈.C₂², C₂⁴.₁₇₈D₄

Smallest permutation representation of C₂⁴.₁₇₈D₄
►On 32 points

Generators in S₃₂

(2 30)(4 32)(6 26)(8 28)(10 18)(12 20)(14 22)(16 24)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	8A	8B	8C	8D
order	1	2	2	2	2	···	2	2	4	4	4	4	4	···	4	4	4	4	8	8	8	8
size	1	1	1	1	2	···	2	8	2	2	2	2	4	···	4	8	8	8	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₈.C₂²

kernel

C₂⁴.₁₇₈D₄

C₂⁴.₄C₄

C₂³.₃₈D₄

Q₈⋊D₄

C₂²⋊Q₁₆

C₂².₁₉C₂⁴

C₂×C₈.C₂²

Q₈×C₂³

C₂²×C₄

C₂×Q₈

C₂⁴

C₂²

# reps

Matrix representation of C₂⁴.₁₇₈D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
13	16	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

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	9	13	0	1

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	0	0	1	0
0	0	0	0	0	1

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

15	16	0	0	0	0
5	2	0	0	0	0
0	0	0	0	13	0
0	0	15	16	4	9
0	0	0	4	0	0
0	0	11	0	9	1

15	16	0	0	0	0
3	2	0	0	0	0
0	0	0	0	1	0
0	0	8	4	1	15
0	0	1	0	0	0
0	0	8	16	6	13

G:=sub<GL(6,GF(17))| [1,13,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,9,0,0,0,16,0,13,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[15,5,0,0,0,0,16,2,0,0,0,0,0,0,0,15,0,11,0,0,0,16,4,0,0,0,13,4,0,9,0,0,0,9,0,1],[15,3,0,0,0,0,16,2,0,0,0,0,0,0,0,8,1,8,0,0,0,4,0,16,0,0,1,1,0,6,0,0,0,15,0,13] >;

C₂⁴.₁₇₈D₄ in GAP, Magma, Sage, TeX

C_2^4._{178}D_4

% in TeX

G:=Group("C2^4.178D4");

// GroupNames label

G:=SmallGroup(128,1736);

// by ID

G=gap.SmallGroup(128,1736);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=f*b*f=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^3>;

// generators/relations

G = C24.178D4 order 128 = 27

33rd non-split extension by C24 of D4 acting via D4/C22=C2

G = C₂⁴.₁₇₈D₄ order 128 = 2⁷

33^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂²=C₂