C2xD4.3D4 - GroupNames

G = C₂×D₄.₃D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₃D₄

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

Aliases: C₂×D₄.₃D₄, M₄(2).₇C₂³, C₄○D₄.₃₁D₄, D₄.₁₁(C₂×D₄), C₈.₁₂₁(C₂×D₄), (C₂×C₈).₃₇₀D₄, Q₈.₁₁(C₂×D₄), (C₂×D₄).₂₂₃D₄, C₈○D₄⋊₁₁C₂², (C₂×C₄).₁₅C₂⁴, (C₂×Q₈).₁₇₈D₄, (C₂×C₈).₂₆₀C₂³, C₄○D₄.₂₇C₂³, (C₂²×SD₁₆)⋊₃C₂, (C₂×D₄).₇₀C₂³, C₄.₁₆₂(C₂²×D₄), C₈⋊C₂².₃C₂², (C₂×Q₈).₅₈C₂³, C₄.₁₁₃(C₄⋊D₄), C₈.C₄⋊₁₅C₂², (C₂×SD₁₆)⋊₅₆C₂², C₄.D₄⋊₁₂C₂², C₈.C₂²⋊₁₁C₂², C₂³.₃₁₆(C₄○D₄), C₄.₁₀D₄⋊₁₂C₂², C₂².₈₉(C₄⋊D₄), (C₂²×C₄).₉₉₀C₂³, (C₂²×C₈).₂₆₄C₂², (C₂²×D₄).₃₅₂C₂², (C₂×M₄(2)).₅₈C₂², (C₂²×Q₈).₂₈₅C₂², (C₂×C₈○D₄)⋊₇C₂, C₂.₈₆(C₂×C₄⋊D₄), (C₂×C₈.C₄)⋊₂₅C₂, (C₂×C₄).₁₄₂₇(C₂×D₄), (C₂×C₄.D₄)⋊₁₀C₂, (C₂×C₈⋊C₂²).₁₂C₂, (C₂×C₈.C₂²)⋊₂₀C₂, C₂².₁₈(C₂×C₄○D₄), (C₂×C₄).₂₈₉(C₄○D₄), (C₂×C₄.₁₀D₄)⋊₁₀C₂, (C₂×C₄○D₄).₃₀₄C₂², SmallGroup(128,1796)

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

Derived series

C₁ — C₂×C₄ — C₂×D₄.₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×C₈○D₄ — C₂×D₄.₃D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×D₄.₃D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×D₄.₃D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×D₄.₃D₄

Generators and relations for C₂×D₄.₃D₄
G = < a,b,c,d,e | a²=b⁴=c²=1, d⁴=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b^-1, bd=db, cd=dc, ece^-1=bc, ede^-1=d³ >

Subgroups: 460 in 234 conjugacy classes, 100 normal (36 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₄.D₄, C₄.₁₀D₄, C₈.C₄, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₈○D₄, C₈○D₄, C₂×D₈, C₂×SD₁₆, C₂×SD₁₆, C₂×Q₁₆, C₈⋊C₂², C₈⋊C₂², C₈.C₂², C₈.C₂², C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄.D₄, C₂×C₄.₁₀D₄, C₂×C₈.C₄, D₄.₃D₄, C₂×C₈○D₄, C₂²×SD₁₆, C₂×C₈⋊C₂², C₂×C₈.C₂², C₂×D₄.₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, D₄.₃D₄, C₂×C₄⋊D₄, C₂×D₄.₃D₄

Smallest permutation representation of C₂×D₄.₃D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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	···	8J	8K	8L	8M	8N
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	8	8	8	8
size	1	1	1	1	2	2	4	4	8	8	2	2	2	2	4	4	8	8	2	2	2	2	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄.₃D₄

kernel

C₂×D₄.₃D₄

C₂×C₄.D₄

C₂×C₄.₁₀D₄

C₂×C₈.C₄

D₄.₃D₄

C₂×C₈○D₄

C₂²×SD₁₆

C₂×C₈⋊C₂²

C₂×C₈.C₂²

C₂×C₈

C₂×D₄

C₂×Q₈

C₄○D₄

C₂×C₄

C₂³

C₂

# reps

Matrix representation of C₂×D₄.₃D₄ ►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	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	15	0	0
0	0	1	16	0	0
0	0	0	0	16	2
0	0	0	0	16	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	15
0	0	0	0	1	16
0	0	16	2	0	0
0	0	16	1	0	0

0	4	0	0	0	0
4	0	0	0	0	0
0	0	0	10	0	0
0	0	12	10	0	0
0	0	0	0	0	10
0	0	0	0	12	10

0	13	0	0	0	0
4	0	0	0	0	0
0	0	0	10	0	0
0	0	5	0	0	0
0	0	0	0	10	7
0	0	0	0	5	7

G:=sub<GL(6,GF(17))| [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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,1,0,0,0,0,15,16,0,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,0,0,0,0,12,0,0,0,0,10,10],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7] >;

C₂×D₄.₃D₄ in GAP, Magma, Sage, TeX

C_2\times D_4._3D_4

% in TeX

G:=Group("C2xD4.3D4");

// GroupNames label

G:=SmallGroup(128,1796);

// by ID

G=gap.SmallGroup(128,1796);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2804,172,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C2×D4.3D4 order 128 = 27

Direct product of C2 and D4.3D4

G = C₂×D₄.₃D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₃D₄