C2xD4oSD16 - GroupNames

G = C₂xD₄oSD₁₆ order 128 = 2⁷

Direct product of C₂ and D₄oSD₁₆

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

Aliases: C₂xD₄oSD₁₆, D₈:₅C₂³, C₈.₄C₂⁴, C₄.₉C₂⁵, Q₁₆:₅C₂³, D₄.₆C₂⁴, Q₈.₆C₂⁴, SD₁₆:₇C₂³, M₄(2):₈C₂³, 2₊¹⁺⁴:₉C₂², 2_-¹⁺⁴:₇C₂², Q₈ o(C₂xSD₁₆), SD₁₆ o(C₂xQ₈), D₄ o(C₂xSD₁₆), SD₁₆ o(C₂xD₄), (C₂xC₈):₇C₂³, C₄oD₄.₃₉D₄, D₄.₆₂(C₂xD₄), C₄oD₄:₂C₂³, Q₈.₆₄(C₂xD₄), (C₂xD₄).₃₅₈D₄, C₄oD₈:₁₀C₂², C₈oD₄:₁₄C₂², (C₂xD₈):₅₇C₂², (C₂xQ₈).₂₇₇D₄, (C₂xQ₈):₁₁C₂³, C₂.₄₄(D₄xC₂³), C₈:C₂²:₁₃C₂², (C₂xC₄).₆₁₅C₂⁴, (C₂²xC₈):₂₉C₂², (C₂xQ₁₆):₆₀C₂², C₄.₁₂₆(C₂²xD₄), C₂³.₄₈₆(C₂xD₄), (C₂xSD₁₆):₆₃C₂², (C₂²xSD₁₆):₁₀C₂, (C₂xD₄).₃₄₆C₂³, C₈.C₂²:₁₃C₂², (C₂²xQ₈):₄₈C₂², C₂².₁₈(C₂²xD₄), (C₂xM₄(2)):₆₀C₂², (C₂x2₊¹⁺⁴):₁₄C₂, (C₂x2_-¹⁺⁴):₁₁C₂, (C₂²xC₄).₁₂₂₆C₂³, (C₂²xD₄).₄₄₃C₂², (C₂xQ₈)o(C₂xSD₁₆), (C₂xC₈oD₄):₁₁C₂, (C₂xC₄oD₈):₃₁C₂, (C₂xC₈:C₂²):₃₅C₂, (C₂xC₄).₁₁₁₄(C₂xD₄), (C₂xC₄oD₄):₅₇C₂², (C₂xC₈.C₂²):₃₄C₂, SmallGroup(128,2314)

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

Derived series

C₁ — C₄ — C₂xD₄oSD₁₆

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂xC₄oD₄ — C₂x2₊¹⁺⁴ — C₂xD₄oSD₁₆

Lower central

C₁ — C₂ — C₄ — C₂xD₄oSD₁₆

Upper central

C₁ — C₂² — C₂xC₄oD₄ — C₂xD₄oSD₁₆

Jennings

C₁ — C₂ — C₂ — C₄ — C₂xD₄oSD₁₆

Generators and relations for C₂xD₄oSD₁₆
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d³ >

Subgroups: 1172 in 730 conjugacy classes, 428 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₂²xC₈, C₂xM₄(2), C₈oD₄, C₂xD₈, C₂xSD₁₆, C₂xSD₁₆, C₂xQ₁₆, C₄oD₈, C₈:C₂², C₈.C₂², C₂²xD₄, C₂²xD₄, C₂²xQ₈, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄oD₄, C₂xC₄oD₄, 2₊¹⁺⁴, 2₊¹⁺⁴, 2_-¹⁺⁴, 2_-¹⁺⁴, C₂xC₈oD₄, C₂²xSD₁₆, C₂xC₄oD₈, C₂xC₈:C₂², C₂xC₈.C₂², D₄oSD₁₆, C₂x2₊¹⁺⁴, C₂x2_-¹⁺⁴, C₂xD₄oSD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₂²xD₄, C₂⁵, D₄oSD₁₆, D₄xC₂³, C₂xD₄oSD₁₆

Smallest permutation representation of C₂xD₄oSD₁₆
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim

type

image

C₁

C₂

D₄

D₄oSD₁₆

kernel

C₂xD₄oSD₁₆

C₂xC₈oD₄

C₂²xSD₁₆

C₂xC₄oD₈

C₂xC₈:C₂²

C₂xC₈.C₂²

D₄oSD₁₆

C₂x2₊¹⁺⁴

C₂x2_-¹⁺⁴

C₂xD₄

C₂xQ₈

C₄oD₄

C₂

# reps

Matrix representation of C₂xD₄oSD₁₆ ►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	16	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0

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

11	4	0	0	0	0
12	6	0	0	0	0
0	0	5	12	0	0
0	0	5	5	0	0
0	0	0	0	5	12
0	0	0	0	5	5

16	0	0	0	0	0
14	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	16
0	0	1	0	0	0
0	0	0	16	0	0

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[11,12,0,0,0,0,4,6,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[16,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0] >;

C₂xD₄oSD₁₆ in GAP, Magma, Sage, TeX

C_2\times D_4\circ {\rm SD}_{16}

% in TeX

G:=Group("C2xD4oSD16");

// GroupNames label

G:=SmallGroup(128,2314);

// by ID

G=gap.SmallGroup(128,2314);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,521,4037,2028,124]);

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

// generators/relations

G = C2xD4oSD16 order 128 = 27

Direct product of C2 and D4oSD16

G = C₂xD₄oSD₁₆ order 128 = 2⁷

Direct product of C₂ and D₄oSD₁₆