C2xD4.10D4 - GroupNames

G = C₂×D₄.₁₀D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₁₀D₄

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

Aliases: C₂×D₄.₁₀D₄, C₄².₃₁₂C₂³, M₄(2).₂C₂³, 2_-¹⁺⁴.₆C₂², C₄○D₄.₁₇D₄, D₄.₅₁(C₂×D₄), (C₂×C₄).₉C₂⁴, C₄⋊Q₈⋊₅₀C₂², C₄≀C₂⋊₁₀C₂², Q₈.₅₁(C₂×D₄), C₄.₄₆C₂²≀C₂, (C₂×D₄).₂₉₉D₄, (C₂×Q₈).₂₃₄D₄, C₄○D₄.₄C₂³, C₄.₅₄(C₂²×D₄), (C₂²×C₄).₁₀₉D₄, C₂³.₆₅₁(C₂×D₄), C₈.C₂²⋊₆C₂², (C₂×Q₈).₂₆C₂³, C₄.₁₀D₄⋊₇C₂², C₂².₃₃(C₂²×D₄), C₂².₁₂₁C₂²≀C₂, (C₂×C₄²).₈₀₁C₂², (C₂²×C₄).₉₆₈C₂³, (C₂×2_-¹⁺⁴).₇C₂, (C₂×M₄(2)).₄₅C₂², (C₂²×Q₈).₂₆₅C₂², (C₂×C₄≀C₂)⋊₇C₂, (C₂×C₄⋊Q₈)⋊₂₈C₂, (C₂×C₄).₂₄(C₂×D₄), C₂.₅₄(C₂×C₂²≀C₂), (C₂×C₄.₁₀D₄)⋊₈C₂, (C₂×C₈.C₂²)⋊₁₂C₂, (C₂×C₄○D₄).₁₀₇C₂², SmallGroup(128,1749)

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

Derived series

C₁ — C₂×C₄ — C₂×D₄.₁₀D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂×2_-¹⁺⁴ — C₂×D₄.₁₀D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×D₄.₁₀D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×D₄.₁₀D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×D₄.₁₀D₄

Generators and relations for C₂×D₄.₁₀D₄
G = < a,b,c,d,e | a²=b⁴=c²=1, d⁴=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=ebe^-1=b^-1, dcd^-1=bc, ece^-1=b^-1c, ede^-1=d³ >

Subgroups: 628 in 354 conjugacy classes, 108 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₄.₁₀D₄, C₄≀C₂, C₂×C₄², C₂×C₄⋊C₄, C₄⋊Q₈, C₄⋊Q₈, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₈.C₂², C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, 2_-¹⁺⁴, C₂×C₄.₁₀D₄, C₂×C₄≀C₂, D₄.₁₀D₄, C₂×C₄⋊Q₈, C₂×C₈.C₂², C₂×2_-¹⁺⁴, C₂×D₄.₁₀D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, D₄.₁₀D₄, C₂×C₂²≀C₂, C₂×D₄.₁₀D₄

Smallest permutation representation of C₂×D₄.₁₀D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₄.₁₀D₄

kernel

C₂×D₄.₁₀D₄

C₂×C₄.₁₀D₄

C₂×C₄≀C₂

D₄.₁₀D₄

C₂×C₄⋊Q₈

C₂×C₈.C₂²

C₂×2_-¹⁺⁴

C₂²×C₄

C₂×D₄

C₂×Q₈

C₄○D₄

C₂

# reps

Matrix representation of C₂×D₄.₁₀D₄ ►in 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

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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	10	10	8	14
0	0	16	16	11	2
0	0	16	7	0	0
0	0	10	9	1	8

0	16	0	0	0	0
1	0	0	0	0	0
0	0	16	16	11	2
0	0	7	7	9	3
0	0	1	10	0	0
0	0	6	7	10	11

0	1	0	0	0	0
1	0	0	0	0	0
0	0	10	16	0	0
0	0	16	7	0	0
0	0	10	10	8	14
0	0	0	7	16	9

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,16,0,0,1,0,1,0,0,0,0,0,16,1,0,0,0,0,15,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,10,16,16,10,0,0,10,16,7,9,0,0,8,11,0,1,0,0,14,2,0,8],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,7,1,6,0,0,16,7,10,7,0,0,11,9,0,10,0,0,2,3,0,11],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,16,10,0,0,0,16,7,10,7,0,0,0,0,8,16,0,0,0,0,14,9] >;

C₂×D₄.₁₀D₄ in GAP, Magma, Sage, TeX

C_2\times D_4._{10}D_4

% in TeX

G:=Group("C2xD4.10D4");

// GroupNames label

G:=SmallGroup(128,1749);

// by ID

G=gap.SmallGroup(128,1749);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,2804,1411,718,172,2028]);

// 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=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;

// generators/relations

G = C2×D4.10D4 order 128 = 27

Direct product of C2 and D4.10D4

G = C₂×D₄.₁₀D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₁₀D₄