(C2xQ8):16D4 - GroupNames

G = (C₂xQ₈):₁₆D₄ order 128 = 2⁷

12^nd semidirect product of C₂xQ₈ and D₄ acting via D₄/C₂=C₂²

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

Aliases: C₄oD₄:₂D₄, (C₂xQ₈):₁₆D₄, Q₈:D₄:₃C₂, D₄.₄₅(C₂xD₄), Q₈.₄₅(C₂xD₄), D₄:D₄:₁₆C₂, C₄.₄₁C₂²wrC₂, (C₂xD₄).₂₉₄D₄, C₄:C₄.₁₂C₂³, C₂²:C₈:₉C₂², C₄.₄₇(C₂²xD₄), (C₂xC₄).₂₂₉C₂⁴, (C₂xC₈).₃₀₁C₂³, C₂.₉(D₄oSD₁₆), (C₂xD₄).₃₀C₂³, (C₂xD₈).₅₂C₂², C₂³.₂₃₃(C₂xD₄), D₄:C₄:₇₂C₂², C₂².₂₉C₂⁴:₄C₂, Q₈:C₄:₁₅C₂², C₂².₂₁C₂²wrC₂, (C₂²xSD₁₆):₂₀C₂, C₂³.₃₈D₄:₄C₂, C₄:D₄.₁₇C₂², (C₂x2_-¹⁺⁴):₁C₂, (C₂xQ₈).₃₅₅C₂³, (C₂²xQ₈):₁₅C₂², C₂³.₂₄D₄:₂₂C₂, (C₂²xC₄).₂₇₇C₂³, (C₂²xC₈).₃₃₆C₂², C₂².₄₈₉(C₂²xD₄), C₄²:C₂.₉₇C₂², (C₂xSD₁₆).₁₂₉C₂², (C₂²xD₄).₃₂₇C₂², (C₂xM₄(2)).₄₁C₂², (C₂xC₈:C₂²):₉C₂, (C₂xC₄).₄₅₆(C₂xD₄), (C₂xC₄oD₄):₄C₂², (C₂²xC₈):C₂:₅C₂, C₂.₄₇(C₂xC₂²wrC₂), SmallGroup(128,1742)

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

Derived series

C₁ — C₂xC₄ — (C₂xQ₈):₁₆D₄

Chief series

C₁ — C₂ — C₂² — C₂xC₄ — C₂²xC₄ — C₂xC₄oD₄ — C₂x2_-¹⁺⁴ — (C₂xQ₈):₁₆D₄

Lower central

C₁ — C₂ — C₂xC₄ — (C₂xQ₈):₁₆D₄

Upper central

C₁ — C₂² — C₂xC₄oD₄ — (C₂xQ₈):₁₆D₄

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — (C₂xQ₈):₁₆D₄

Generators and relations for (C₂xQ₈):₁₆D₄
G = < a,b,c,d,e | a²=b⁴=d⁴=e²=1, c²=b², ab=ba, ac=ca, dad^-1=ab², ae=ea, cbc^-1=b^-1, dbd^-1=abc, ebe=ab^-1c, cd=dc, ece=b²c, ede=d^-1 >

Subgroups: 732 in 366 conjugacy classes, 108 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₂²:C₈, D₄:C₄, Q₈:C₄, C₄²:C₂, C₂²wrC₂, C₄:D₄, C₄.₄D₄, C₄:₁D₄, C₂²xC₈, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂²xQ₈, C₂²xQ₈, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄oD₄, C₂xC₄oD₄, 2_-¹⁺⁴, (C₂²xC₈):C₂, C₂³.₂₄D₄, C₂³.₃₈D₄, Q₈:D₄, D₄:D₄, C₂².₂₉C₂⁴, C₂²xSD₁₆, C₂xC₈:C₂², C₂x2_-¹⁺⁴, (C₂xQ₈):₁₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₂²wrC₂, C₂²xD₄, C₂xC₂²wrC₂, D₄oSD₁₆, (C₂xQ₈):₁₆D₄

Smallest permutation representation of (C₂xQ₈):₁₆D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₄oSD₁₆

kernel

(C₂xQ₈):₁₆D₄

(C₂²xC₈):C₂

C₂³.₂₄D₄

C₂³.₃₈D₄

Q₈:D₄

D₄:D₄

C₂².₂₉C₂⁴

C₂²xSD₁₆

C₂xC₈:C₂²

C₂x2_-¹⁺⁴

C₂xD₄

C₂xQ₈

C₄oD₄

C₂

# reps

Matrix representation of (C₂xQ₈):₁₆D₄ ►in GL₆(F₁₇)

1	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	16	0
0	0	0	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	12	12	0	0
0	0	12	5	0	0
0	0	0	0	5	5
0	0	0	0	5	12

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

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

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

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

(C₂xQ₈):₁₆D₄ in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_{16}D_4

% in TeX

G:=Group("(C2xQ8):16D4");

// GroupNames label

G:=SmallGroup(128,1742);

// by ID

G=gap.SmallGroup(128,1742);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,2804,1411,172]);

// Polycyclic

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

// generators/relations

G = (C2xQ8):16D4 order 128 = 27

12nd semidirect product of C2xQ8 and D4 acting via D4/C2=C22

G = (C₂xQ₈):₁₆D₄ order 128 = 2⁷

12^nd semidirect product of C₂xQ₈ and D₄ acting via D₄/C₂=C₂²