C2xD4oSD16 - GroupNames

G = C₂×D₄○SD₁₆ order 128 = 2⁷

Direct product of C₂ and D₄○SD₁₆

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

Aliases: C₂×D₄○SD₁₆, 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 ○(C₂×SD₁₆), SD₁₆ ○(C₂×Q₈), D₄ ○(C₂×SD₁₆), SD₁₆ ○(C₂×D₄), (C₂×C₈)⋊₇C₂³, C₄○D₄.₃₉D₄, D₄.₆₂(C₂×D₄), C₄○D₄⋊₂C₂³, Q₈.₆₄(C₂×D₄), (C₂×D₄).₃₅₈D₄, C₄○D₈⋊₁₀C₂², C₈○D₄⋊₁₄C₂², (C₂×D₈)⋊₅₇C₂², (C₂×Q₈).₂₇₇D₄, (C₂×Q₈)⋊₁₁C₂³, C₂.₄₄(D₄×C₂³), C₈⋊C₂²⋊₁₃C₂², (C₂×C₄).₆₁₅C₂⁴, (C₂²×C₈)⋊₂₉C₂², (C₂×Q₁₆)⋊₆₀C₂², C₄.₁₂₆(C₂²×D₄), C₂³.₄₈₆(C₂×D₄), (C₂×SD₁₆)⋊₆₃C₂², (C₂²×SD₁₆)⋊₁₀C₂, (C₂×D₄).₃₄₆C₂³, C₈.C₂²⋊₁₃C₂², (C₂²×Q₈)⋊₄₈C₂², C₂².₁₈(C₂²×D₄), (C₂×M₄(2))⋊₆₀C₂², (C₂×2₊¹⁺⁴)⋊₁₄C₂, (C₂×2_-¹⁺⁴)⋊₁₁C₂, (C₂²×C₄).₁₂₂₆C₂³, (C₂²×D₄).₄₄₃C₂², (C₂×Q₈)○(C₂×SD₁₆), (C₂×C₈○D₄)⋊₁₁C₂, (C₂×C₄○D₈)⋊₃₁C₂, (C₂×C₈⋊C₂²)⋊₃₅C₂, (C₂×C₄).₁₁₁₄(C₂×D₄), (C₂×C₄○D₄)⋊₅₇C₂², (C₂×C₈.C₂²)⋊₃₄C₂, SmallGroup(128,2314)

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

Derived series

C₁ — C₄ — C₂×D₄○SD₁₆

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×2₊¹⁺⁴ — C₂×D₄○SD₁₆

Lower central

C₁ — C₂ — C₄ — C₂×D₄○SD₁₆

Upper central

C₁ — C₂² — C₂×C₄○D₄ — C₂×D₄○SD₁₆

Jennings

C₁ — C₂ — C₂ — C₄ — C₂×D₄○SD₁₆

Generators and relations for C₂×D₄○SD₁₆
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, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂×C₈, C₂×C₈, 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, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂×D₈, C₂×SD₁₆, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, 2_-¹⁺⁴, 2_-¹⁺⁴, C₂×C₈○D₄, C₂²×SD₁₆, C₂×C₄○D₈, C₂×C₈⋊C₂², C₂×C₈.C₂², D₄○SD₁₆, C₂×2₊¹⁺⁴, C₂×2_-¹⁺⁴, C₂×D₄○SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₄○SD₁₆, D₄×C₂³, C₂×D₄○SD₁₆

Smallest permutation representation of C₂×D₄○SD₁₆
►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

D₄○SD₁₆

kernel

C₂×D₄○SD₁₆

C₂×C₈○D₄

C₂²×SD₁₆

C₂×C₄○D₈

C₂×C₈⋊C₂²

C₂×C₈.C₂²

D₄○SD₁₆

C₂×2₊¹⁺⁴

C₂×2_-¹⁺⁴

C₂×D₄

C₂×Q₈

C₄○D₄

C₂

# reps

Matrix representation of C₂×D₄○SD₁₆ ►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] >;

C₂×D₄○SD₁₆ 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 = C2×D4○SD16 order 128 = 27

Direct product of C2 and D4○SD16

G = C₂×D₄○SD₁₆ order 128 = 2⁷

Direct product of C₂ and D₄○SD₁₆