C4:SD16 - GroupNames

G = C₄⋊SD₁₆ order 64 = 2⁶

The semidirect product of C₄ and SD₁₆ acting via SD₁₆/Q₈=C₂

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

Aliases: Q₈⋊₁D₄, C₄⋊₃SD₁₆, C₄².₁₇C₂², C₄⋊C₈⋊₈C₂, (C₄×Q₈)⋊₃C₂, C₄.₃₀(C₂×D₄), (C₂×C₄).₁₃₀D₄, C₄⋊₁D₄.₃C₂, D₄⋊C₄⋊₁₀C₂, C₂.₇(C₂×SD₁₆), (C₂×SD₁₆)⋊₁₁C₂, C₄.₄₀(C₄○D₄), C₄⋊C₄.₅₇C₂², (C₂×C₈).₂₉C₂², (C₂×C₄).₈₈C₂³, C₂².₈₄(C₂×D₄), C₂.₁₂(C₄⋊D₄), C₂.₁₀(C₈⋊C₂²), (C₂×D₄).₁₂C₂², (C₂×Q₈).₅₀C₂², SmallGroup(64,141)

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

Derived series

C₁ — C₂×C₄ — C₄⋊SD₁₆

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×Q₈ — C₄×Q₈ — C₄⋊SD₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₄⋊SD₁₆

Upper central

C₁ — C₂² — C₄² — C₄⋊SD₁₆

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄⋊SD₁₆

Generators and relations for C₄⋊SD₁₆
G = < a,b,c | a⁴=b⁸=c²=1, bab^-1=cac=a^-1, cbc=b³ >

Subgroups: 133 in 64 conjugacy classes, 29 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂×D₄, C₂×D₄, C₂×Q₈, D₄⋊C₄, C₄⋊C₈, C₄×Q₈, C₄⋊₁D₄, C₂×SD₁₆, C₄⋊SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₈⋊C₂², C₄⋊SD₁₆

Character table of C₄⋊SD₁₆

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D
size	1	1	1	1	8	8	2	2	2	2	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	linear of order 2
ρ₃	1	1	1	1	1	-1	-1	1	-1	1	-1	-1	1	1	-1	-1	1	-1	1	linear of order 2
ρ₄	1	1	1	1	-1	1	-1	1	-1	1	1	1	-1	-1	-1	-1	1	-1	1	linear of order 2
ρ₅	1	1	1	1	1	-1	-1	1	-1	1	1	1	-1	-1	-1	1	-1	1	-1	linear of order 2
ρ₆	1	1	1	1	-1	1	-1	1	-1	1	-1	-1	1	1	-1	1	-1	1	-1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	-1	-1	-1	-1	linear of order 2
ρ₈	1	1	1	1	-1	-1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₉	2	2	2	2	0	0	-2	-2	-2	-2	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	-2	2	-2	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	0	0	0	-2	0	2	0	0	-2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	2	-2	0	0	0	-2	0	2	0	0	2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	-2	2	-2	0	0	0	2	0	-2	-2i	2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₄	2	-2	2	-2	0	0	0	2	0	-2	2i	-2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₅	2	-2	-2	2	0	0	2	0	-2	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₁₆	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₇	2	-2	-2	2	0	0	2	0	-2	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₉	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²

Smallest permutation representation of C₄⋊SD₁₆
►On 32 points

Generators in S₃₂

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

(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)

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

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

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

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

C₄⋊SD₁₆ is a maximal subgroup of
C₄².₄₄₃D₄ C₄².₄₄₄D₄ C₄².₂₂₃D₄ C₄².₄₅₀D₄ C₄².₂₃₀D₄ C₄².₂₃₃D₄ C₄².₂₇₀D₄ C₄².₂₇₄D₄ C₄².₂₉₄D₄ C₄².₃₀₂D₄
Q₈⋊D_4p: Q₈⋊D₈ Q₈⋊₃D₈ C₈⋊₁₃SD₁₆ Q₈⋊₂D₁₂ Q₈⋊D₁₂ Q₈⋊D₂₀ Q₈⋊D₂₈ ...
C_4p⋊SD₁₆: C₈⋊₁₄SD₁₆ C₈⋊SD₁₆ C₈⋊₂SD₁₆ Dic₆⋊₈D₄ Dic₆⋊₉D₄ Dic₁₀⋊₈D₄ Dic₁₀⋊₉D₄ Dic₁₄⋊₈D₄ ...
(C_p×Q₈)⋊D₄: C₄².₂₁₂D₄ C₄².₁₆C₂³ Dic₃⋊₅SD₁₆ Dic₅⋊₅SD₁₆ Dic₇⋊₅SD₁₆ ...
C₈⋊_pD₄⋊C₂: D₄⋊SD₁₆ C₄².₂₆₆D₄ C₄².₂₇₅D₄ C₄².₄₀₈C₂³ C₄².₄₁₀C₂³ C₄².₂₉₅D₄ C₄².₂₉₉D₄ C₄.2_-¹⁺⁴ ...
C₄⋊C₄.D_2p: C₄².₁₈₁C₂³ Q₈⋊SD₁₆ Q₈⋊₆SD₁₆ C₄².₁₈₉C₂³ Q₈.₂SD₁₆ Q₈.₂D₈ C₄².₂₄₉C₂³ C₄².₂₅₃C₂³ ...
C₄⋊SD₁₆ is a maximal quotient of
C₄².₉₉D₄ C₄².₁₁₈D₄ C₄⋊C₄⋊₇D₄ (C₂×C₄)⋊₅SD₁₆ (C₂×C₄).₁₉Q₁₆
Q₈⋊D_4p: C₈⋊₁₃SD₁₆ Q₈⋊₂D₁₂ Q₈⋊D₂₀ Q₈⋊D₂₈ ...
C_4p⋊SD₁₆: C₈⋊₁₄SD₁₆ C₈⋊SD₁₆ C₈⋊₂SD₁₆ Dic₆⋊₈D₄ Dic₆⋊₉D₄ Dic₁₀⋊₈D₄ Dic₁₀⋊₉D₄ Dic₁₄⋊₈D₄ ...
(C_p×Q₈)⋊D₄: (C₂×C₈)⋊₂₀D₄ Dic₃⋊₅SD₁₆ Dic₅⋊₅SD₁₆ Dic₇⋊₅SD₁₆ ...
C₄⋊C₄.D_2p: Q₈⋊₁Q₁₆ C₈.SD₁₆ (C₂×SD₁₆)⋊₁₅C₄ C₄.₆₇(C₄×D₄) C₄².₃₀Q₈ C₄⋊C₄.₁₀₆D₄ C₂.(C₈⋊₃Q₈) Dic₆⋊₂D₄ ...

Matrix representation of C₄⋊SD₁₆ ►in GL₄(𝔽₁₇) generated by

16	0	0	0
0	16	0	0
0	0	13	8
0	0	0	4

12	12	0	0
5	12	0	0
0	0	1	15
0	0	1	16

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

G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,8,4],[12,5,0,0,12,12,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C₄⋊SD₁₆ in GAP, Magma, Sage, TeX

C_4\rtimes {\rm SD}_{16}

% in TeX

G:=Group("C4:SD16");

// GroupNames label

G:=SmallGroup(64,141);

// by ID

G=gap.SmallGroup(64,141);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,158,1444,376,88]);

// Polycyclic

G:=Group<a,b,c|a^4=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^3>;

// generators/relations

Export

Character table of C₄⋊SD₁₆ in TeX

G = C4⋊SD16 order 64 = 26

The semidirect product of C4 and SD16 acting via SD16/Q8=C2

G = C₄⋊SD₁₆ order 64 = 2⁶

The semidirect product of C₄ and SD₁₆ acting via SD₁₆/Q₈=C₂