C2xD4.4D4

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

Aliases: C₂×D₄.₄D₄, M₄(2).₈C₂³, C₄○D₄.₃₂D₄, D₄.₁₂(C₂×D₄), C₈.₁₁₂(C₂×D₄), (C₂×C₈).₃₅₁D₄, Q₈.₁₂(C₂×D₄), (C₂×D₄).₂₂₄D₄, C₈○D₄⋊₁₂C₂², (C₂²×D₈)⋊₁₇C₂, (C₂×D₈)⋊₄₈C₂², (C₂×C₄).₁₆C₂⁴, (C₂×Q₈).₁₇₉D₄, C₈⋊C₂²⋊₁₀C₂², (C₂×C₈).₂₆₁C₂³, C₄○D₄.₂₈C₂³, (C₂×D₄).₇₁C₂³, C₄.₁₆₃(C₂²×D₄), C₄.₁₁₄(C₄⋊D₄), C₈.C₄⋊₁₆C₂², C₄.D₄⋊₁₃C₂², C₂³.₃₁₇(C₄○D₄), C₂².₉₀(C₄⋊D₄), (C₂²×C₄).₉₉₁C₂³, (C₂²×C₈).₂₆₅C₂², (C₂²×D₄).₃₅₃C₂², (C₂×M₄(2)).₅₉C₂², (C₂×C₈○D₄)⋊₈C₂, (C₂×C₈⋊C₂²)⋊₂₀C₂, C₂.₈₇(C₂×C₄⋊D₄), (C₂×C₈.C₄)⋊₂₆C₂, (C₂×C₄.D₄)⋊₁₁C₂, (C₂×C₄).₁₄₂₈(C₂×D₄), C₂².₁₉(C₂×C₄○D₄), (C₂×C₄).₂₉₀(C₄○D₄), (C₂×C₄○D₄).₃₀₅C₂², SmallGroup(128,1797)

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

Derived series

C₁ — C₂×C₄ — C₂×D₄.₄D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×C₈○D₄ — C₂×D₄.₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×D₄.₄D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×D₄.₄D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×D₄.₄D₄

Subgroups: 540 in 244 conjugacy classes, 100 normal (28 characteristic)
C₁, C₂, C₂^[×2], C₂^[×8], C₄^[×4], C₄^[×2], C₂²^[×3], C₂²^[×22], C₈^[×4], C₈^[×6], C₂×C₄^[×6], C₂×C₄^[×5], D₄^[×2], D₄^[×17], Q₈^[×2], Q₈, C₂³, C₂³^[×13], C₂×C₈^[×2], C₂×C₈^[×4], C₂×C₈^[×7], M₄(2)^[×6], M₄(2)^[×7], D₈^[×16], SD₁₆^[×8], C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄^[×4], C₂×D₄^[×11], C₂×Q₈, C₄○D₄^[×4], C₄○D₄^[×2], C₂⁴^[×2], C₄.D₄^[×8], C₈.C₄^[×4], C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2)^[×2], C₂×M₄(2), C₈○D₄^[×4], C₈○D₄^[×2], C₂×D₈^[×4], C₂×D₈^[×6], C₂×SD₁₆^[×2], C₈⋊C₂²^[×8], C₈⋊C₂²^[×4], C₂²×D₄^[×2], C₂×C₄○D₄, C₂×C₄.D₄^[×2], C₂×C₈.C₄, D₄.₄D₄^[×8], C₂×C₈○D₄, C₂²×D₈, C₂×C₈⋊C₂²^[×2], C₂×D₄.₄D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×8], C₂³^[×15], C₂×D₄^[×12], C₄○D₄^[×2], C₂⁴, C₄⋊D₄^[×4], C₂²×D₄^[×2], C₂×C₄○D₄, D₄.₄D₄^[×2], C₂×C₄⋊D₄, C₂×D₄.₄D₄

Generators and relations
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, cd=dc, ece=bc, ede=b²d³ >

Smallest permutation representation
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₁₇)

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

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

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

13	8	0	0	0	0
0	4	0	0	0	0
0	0	14	14	0	0
0	0	3	14	0	0
0	0	3	3	0	6
0	0	14	0	14	11

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

G:=sub<GL(6,GF(17))| [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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,16,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,16,0,0,1,1,0,16,0,0,2,0,0,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,3,3,14,0,0,14,14,3,0,0,0,0,0,0,14,0,0,0,0,6,11],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄.₄D₄

kernel

C₂×D₄.₄D₄

C₂×C₄.D₄

C₂×C₈.C₄

D₄.₄D₄

C₂×C₈○D₄

C₂²×D₈

C₂×C₈⋊C₂²

C₂×C₈

C₂×D₄

C₂×Q₈

C₄○D₄

C₂×C₄

C₂³

C₂

# reps

In GAP, Magma, Sage, TeX

C_2\times D_4._4D_4

% in TeX

G:=Group("C2xD4.4D4");

// GroupNames label

G:=SmallGroup(128,1797);

// by ID

G=gap.SmallGroup(128,1797);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C₂×D₄.₄D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₄D₄

G = C2×D4.4D4 order 128 = 27

Direct product of C2 and D4.4D4

G = C₂×D₄.₄D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₄D₄