C4oD4:D4 - GroupNames

G = C₄oD₄:D₄ order 128 = 2⁷

1^st semidirect product of C₄oD₄ and D₄ acting via D₄/C₂=C₂²

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

Aliases: C₄oD₄:₁D₄, (C₂xD₄):₂₀D₄, D₄.₄₃(C₂xD₄), C₂.₆(D₄oD₈), (C₂²xD₈):₇C₂, Q₈.₄₃(C₂xD₄), D₄:D₄:₁₅C₂, C₂²:D₈:₁₃C₂, C₄.₃₉C₂²wrC₂, (C₂xD₈):₃₈C₂², C₄:C₄.₁₀C₂³, C₄:D₄:₁C₂², C₂²:C₈:₇C₂², (C₂xQ₈).₂₂₉D₄, (C₂²xC₈):₈C₂², C₄.₄₅(C₂²xD₄), (C₂xC₄).₂₂₇C₂⁴, (C₂xC₈).₁₂₉C₂³, (C₂xSD₁₆):₆C₂², (C₂xD₄).₂₉C₂³, C₂³.₂₃₁(C₂xD₄), D₄:C₄:₁₂C₂², C₂².₂₉C₂⁴:₃C₂, C₄²:C₂:₇C₂², Q₈:C₄:₆₆C₂², C₂².₁₉C₂²wrC₂, C₂³.₃₇D₄:₄C₂, C₂³.₂₄D₄:₆C₂, (C₂²xD₄):₁₈C₂², (C₂xM₄(2)):₄C₂², (C₂x2₊¹⁺⁴):₁C₂, (C₂xQ₈).₃₅₄C₂³, (C₂²xC₄).₂₇₅C₂³, C₂².₄₈₇(C₂²xD₄), (C₂xC₈:C₂²):₈C₂, (C₂xC₄).₄₅₄(C₂xD₄), (C₂xC₄oD₄):₃C₂², (C₂²xC₈):C₂:₃C₂, C₂.₄₅(C₂xC₂²wrC₂), SmallGroup(128,1740)

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

Derived series

C₁ — C₂xC₄ — C₄oD₄:D₄

Chief series

C₁ — C₂ — C₂² — C₂xC₄ — C₂²xC₄ — C₂xC₄oD₄ — C₂x2₊¹⁺⁴ — C₄oD₄:D₄

Lower central

C₁ — C₂ — C₂xC₄ — C₄oD₄:D₄

Upper central

C₁ — C₂² — C₂xC₄oD₄ — C₄oD₄:D₄

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₄oD₄:D₄

Generators and relations for C₄oD₄:D₄
G = < a,b,c,d,e | a⁴=c²=d⁴=e²=1, b²=a², ab=ba, ac=ca, ad=da, eae=a^-1, cbc=dbd^-1=ebe=a²b, dcd^-1=ece=bc, ede=d^-1 >

Subgroups: 908 in 392 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, C₄oD₄, C₄oD₄, C₂⁴, C₂²:C₈, D₄:C₄, Q₈:C₄, C₄²:C₂, C₂²wrC₂, C₄:D₄, C₄.₄D₄, C₄:₁D₄, C₂²xC₈, C₂xM₄(2), C₂xD₈, C₂xD₈, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂²xD₄, C₂²xD₄, C₂xC₄oD₄, C₂xC₄oD₄, C₂xC₄oD₄, 2₊¹⁺⁴, (C₂²xC₈):C₂, C₂³.₂₄D₄, C₂³.₃₇D₄, C₂²:D₈, D₄:D₄, C₂².₂₉C₂⁴, C₂²xD₈, C₂xC₈:C₂², C₂x2₊¹⁺⁴, C₄oD₄:D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₂²wrC₂, C₂²xD₄, C₂xC₂²wrC₂, D₄oD₈, C₄oD₄:D₄

Smallest permutation representation of C₄oD₄:D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₄oD₈

kernel

C₄oD₄:D₄

(C₂²xC₈):C₂

C₂³.₂₄D₄

C₂³.₃₇D₄

C₂²:D₈

D₄:D₄

C₂².₂₉C₂⁴

C₂²xD₈

C₂xC₈:C₂²

C₂x2₊¹⁺⁴

C₂xD₄

C₂xQ₈

C₄oD₄

C₂

# reps

Matrix representation of C₄oD₄:D₄ ►in GL₆(F₁₇)

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	1	0	0
0	0	16	0	0	0

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	16
0	0	0	0	1	0

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

0	15	0	0	0	0
9	0	0	0	0	0
0	0	3	3	0	0
0	0	3	14	0	0
0	0	0	0	14	14
0	0	0	0	14	3

0	2	0	0	0	0
9	0	0	0	0	0
0	0	0	0	14	3
0	0	0	0	3	3
0	0	14	3	0	0
0	0	3	3	0	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,14,3,0,0,0,0,3,3,0,0] >;

C₄oD₄:D₄ in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes D_4

% in TeX

G:=Group("C4oD4:D4");

// GroupNames label

G:=SmallGroup(128,1740);

// by ID

G=gap.SmallGroup(128,1740);

# by ID

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

// Polycyclic

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

// generators/relations

G = C4oD4:D4 order 128 = 27

1st semidirect product of C4oD4 and D4 acting via D4/C2=C22

G = C₄oD₄:D₄ order 128 = 2⁷

1^st semidirect product of C₄oD₄ and D₄ acting via D₄/C₂=C₂²