C4oD4:D4 - GroupNames

G = C₄○D₄⋊D₄ order 128 = 2⁷

1^st semidirect product of C₄○D₄ and D₄ acting via D₄/C₂=C₂²

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

Aliases: C₄○D₄⋊₁D₄, (C₂×D₄)⋊₂₀D₄, D₄.₄₃(C₂×D₄), C₂.₆(D₄○D₈), (C₂²×D₈)⋊₇C₂, Q₈.₄₃(C₂×D₄), D₄⋊D₄⋊₁₅C₂, C₂²⋊D₈⋊₁₃C₂, C₄.₃₉C₂²≀C₂, (C₂×D₈)⋊₃₈C₂², C₄⋊C₄.₁₀C₂³, C₄⋊D₄⋊₁C₂², C₂²⋊C₈⋊₇C₂², (C₂×Q₈).₂₂₉D₄, (C₂²×C₈)⋊₈C₂², C₄.₄₅(C₂²×D₄), (C₂×C₄).₂₂₇C₂⁴, (C₂×C₈).₁₂₉C₂³, (C₂×SD₁₆)⋊₆C₂², (C₂×D₄).₂₉C₂³, C₂³.₂₃₁(C₂×D₄), D₄⋊C₄⋊₁₂C₂², C₂².₂₉C₂⁴⋊₃C₂, C₄²⋊C₂⋊₇C₂², Q₈⋊C₄⋊₆₆C₂², C₂².₁₉C₂²≀C₂, C₂³.₃₇D₄⋊₄C₂, C₂³.₂₄D₄⋊₆C₂, (C₂²×D₄)⋊₁₈C₂², (C₂×M₄(2))⋊₄C₂², (C₂×2₊¹⁺⁴)⋊₁C₂, (C₂×Q₈).₃₅₄C₂³, (C₂²×C₄).₂₇₅C₂³, C₂².₄₈₇(C₂²×D₄), (C₂×C₈⋊C₂²)⋊₈C₂, (C₂×C₄).₄₅₄(C₂×D₄), (C₂×C₄○D₄)⋊₃C₂², (C₂²×C₈)⋊C₂⋊₃C₂, C₂.₄₅(C₂×C₂²≀C₂), SmallGroup(128,1740)

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

Derived series

C₁ — C₂×C₄ — C₄○D₄⋊D₄

Chief series

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

Lower central

C₁ — C₂ — C₂×C₄ — C₄○D₄⋊D₄

Upper central

C₁ — C₂² — C₂×C₄○D₄ — C₄○D₄⋊D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄○D₄⋊D₄

Generators and relations for C₄○D₄⋊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, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄²⋊C₂, C₂²≀C₂, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, (C₂²×C₈)⋊C₂, C₂³.₂₄D₄, C₂³.₃₇D₄, C₂²⋊D₈, D₄⋊D₄, C₂².₂₉C₂⁴, C₂²×D₈, C₂×C₈⋊C₂², C₂×2₊¹⁺⁴, C₄○D₄⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₂²≀C₂, D₄○D₈, C₄○D₄⋊D₄

Smallest permutation representation of C₄○D₄⋊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

D₄○D₈

kernel

C₄○D₄⋊D₄

(C₂²×C₈)⋊C₂

C₂³.₂₄D₄

C₂³.₃₇D₄

C₂²⋊D₈

D₄⋊D₄

C₂².₂₉C₂⁴

C₂²×D₈

C₂×C₈⋊C₂²

C₂×2₊¹⁺⁴

C₂×D₄

C₂×Q₈

C₄○D₄

C₂

# reps

Matrix representation of C₄○D₄⋊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] >;

C₄○D₄⋊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 = C4○D4⋊D4 order 128 = 27

1st semidirect product of C4○D4 and D4 acting via D4/C2=C22

G = C₄○D₄⋊D₄ order 128 = 2⁷

1^st semidirect product of C₄○D₄ and D₄ acting via D₄/C₂=C₂²