SD16:C4 - GroupNames

G = SD₁₆⋊C₄ order 64 = 2⁶

1^st semidirect product of SD₁₆ and C₄ acting via C₄/C₂=C₂

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

Aliases: SD₁₆⋊₁C₄, C₄².₉C₂², C₈⋊₂(C₂×C₄), Q₈⋊₂(C₂×C₄), (C₄×Q₈)⋊₂C₂, C₈⋊C₄⋊₁C₂, D₄.₂(C₂×C₄), C₂.D₈⋊₁₁C₂, (C₄×D₄).₅C₂, C₂.₁₅(C₄×D₄), C₄.₄(C₄○D₄), (C₂×C₄).₁₀₁D₄, Q₈⋊C₄⋊₁₆C₂, D₄⋊C₄.₆C₂, C₄⋊C₄.₅₃C₂², C₂.₄(C₈⋊C₂²), (C₂×C₈).₄₉C₂², C₄.₁₂(C₂²×C₄), (C₂×C₄).₇₆C₂³, (C₂×SD₁₆).₁C₂, C₂².₅₄(C₂×D₄), (C₂×D₄).₅₂C₂², C₂.₄(C₈.C₂²), (C₂×Q₈).₄₆C₂², SmallGroup(64,121)

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

Derived series

C₁ — C₄ — SD₁₆⋊C₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — SD₁₆⋊C₄

Lower central

C₁ — C₂ — C₄ — SD₁₆⋊C₄

Upper central

C₁ — C₂² — C₄² — SD₁₆⋊C₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — SD₁₆⋊C₄

Generators and relations for SD₁₆⋊C₄
G = < a,b,c | a⁸=b²=c⁴=1, bab=a³, cac^-1=a⁵, bc=cb >

Subgroups: 101 in 60 conjugacy classes, 35 normal (25 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₈⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₂.D₈, C₄×D₄, C₄×Q₈, C₂×SD₁₆, SD₁₆⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²×C₄, C₂×D₄, C₄○D₄, C₄×D₄, C₈⋊C₂², C₈.C₂², SD₁₆⋊C₄

Character table of SD₁₆⋊C₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	8A	8B	8C	8D
size	1	1	1	1	4	4	2	2	2	2	2	2	4	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	1	1	1	trivial
ρ₂	1	1	1	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	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	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	-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	-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	-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	-1	-1	-1	linear of order 2
ρ₉	1	-1	1	-1	1	-1	1	-1	-i	i	-i	i	-i	-i	i	-1	1	i	i	-1	-i	1	linear of order 4
ρ₁₀	1	-1	1	-1	-1	1	1	-1	i	-i	i	-i	-i	-i	i	1	-1	i	-i	-1	i	1	linear of order 4
ρ₁₁	1	-1	1	-1	-1	1	1	-1	-i	i	-i	i	-i	i	i	-1	1	-i	-i	1	i	-1	linear of order 4
ρ₁₂	1	-1	1	-1	1	-1	1	-1	i	-i	i	-i	-i	i	i	1	-1	-i	i	1	-i	-1	linear of order 4
ρ₁₃	1	-1	1	-1	-1	1	1	-1	i	-i	i	-i	i	-i	-i	-1	1	i	i	1	-i	-1	linear of order 4
ρ₁₄	1	-1	1	-1	1	-1	1	-1	-i	i	-i	i	i	-i	-i	1	-1	i	-i	1	i	-1	linear of order 4
ρ₁₅	1	-1	1	-1	1	-1	1	-1	i	-i	i	-i	i	i	-i	-1	1	-i	-i	-1	i	1	linear of order 4
ρ₁₆	1	-1	1	-1	-1	1	1	-1	-i	i	-i	i	i	i	-i	1	-1	-i	i	-1	-i	1	linear of order 4
ρ₁₇	2	2	2	2	0	0	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	2	0	0	-2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	2	-2	0	0	-2	2	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	2	-2	0	0	-2	2	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₂	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of SD₁₆⋊C₄
►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 32)(2 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)

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

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

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

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

SD₁₆⋊C₄ is a maximal subgroup of
C₄².₃₅₂C₂³ C₄².₃₅₃C₂³ C₄².₃₅₄C₂³ C₄².₃₅₅C₂³ C₄².₃₅₇C₂³ C₄².₃₅₈C₂³ C₄².₃₆₀C₂³ C₄².₃₈₇C₂³ C₄².₃₉₀C₂³ SD₁₆⋊₆D₄ SD₁₆⋊₈D₄ C₄².₄₅C₂³ C₄².₄₆C₂³ C₄².₄₉C₂³ C₄².₅₀C₂³ C₄².₅₅C₂³ C₄².₅₇C₂³ C₄².₆₀C₂³ C₄².₆₂C₂³ C₄².₆₄C₂³ C₄².₅₀₈C₂³ C₄².₅₀₉C₂³ C₄².₅₁₁C₂³ C₄².₅₁₂C₂³ C₄².₅₁₃C₂³ C₄².₅₁₆C₂³ C₄².₅₁₇C₂³ C₄².₅₁₈C₂³ SD₁₆⋊Q₈ SD₁₆⋊₂Q₈ SD₁₆⋊₃Q₈ C₄².₇₅C₂³ C₄².₅₃₂C₂³ GL₂(𝔽₃)⋊C₄
C₄².D_2p: C₄².₃₈₃D₄ C₄×C₈⋊C₂² C₄×C₈.C₂² C₄².₂₂₈D₄ C₄².₂₂₉D₄ C₄².₂₃₀D₄ C₄².₂₃₂D₄ C₄².₂₃₃D₄ ...
C_2p.(C₄×D₄): C₄².₂₇₅C₂³ C₄².₂₇₆C₂³ C₄².₂₇₈C₂³ C₄².₂₈₀C₂³ D₄.S₃⋊C₄ Q₈⋊₃(C₄×S₃) C₂₄⋊C₂⋊C₄ SD₁₆⋊Dic₃ ...
C₈⋊_pD₄⋊C₂: C₄².₃₈₆C₂³ C₄².₃₉₁C₂³ SD₁₆⋊D₄ SD₁₆⋊₇D₄ C₄².₄₃C₂³ C₄².₄₄C₂³ C₄².₅₆C₂³ C₄².₇₂C₂³ ...
SD₁₆⋊C₄ is a maximal quotient of
SD₁₆⋊C₈ D₄.M₄(2) Q₈⋊₂M₄(2) C₈⋊M₄(2) C₈⋊C₄² C₄.Q₈⋊₉C₄ C₄.Q₈⋊₁₀C₄ C₈⋊(C₄⋊C₄)
C₄².D_2p: D₄⋊C₄² Q₈⋊C₄² C₄².₁₆D₆ C₄².₅₁D₆ C₄².₅₆D₆ C₄².₁₆D₁₀ C₄².₅₁D₁₀ C₄².₅₆D₁₀ ...
C_2p.(C₄×D₄): C₂.(C₄×D₈) Q₈⋊(C₄⋊C₄) (C₂×SD₁₆)⋊₁₄C₄ (C₂×SD₁₆)⋊₁₅C₄ D₄⋊C₄⋊C₄ C₂.(C₄×Q₁₆) C₂.(C₈⋊D₄) C₂.(C₈⋊₂D₄) ...

Matrix representation of SD₁₆⋊C₄ ►in GL₆(𝔽₁₇)

0	8	0	0	0	0
2	0	0	0	0	0
0	0	0	8	0	11
0	0	13	8	3	11
0	0	0	11	0	9
0	0	3	11	4	9

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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	16	0	0	0
0	0	0	16	0	0

G:=sub<GL(6,GF(17))| [0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,13,0,3,0,0,8,8,11,11,0,0,0,3,0,4,0,0,11,11,9,9],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

SD₁₆⋊C₄ in GAP, Magma, Sage, TeX

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

% in TeX

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

// GroupNames label

G:=SmallGroup(64,121);

// by ID

G=gap.SmallGroup(64,121);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117]);

// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^3,c*a*c^-1=a^5,b*c=c*b>;

// generators/relations

Export

Character table of SD₁₆⋊C₄ in TeX

G = SD16⋊C4 order 64 = 26

1st semidirect product of SD16 and C4 acting via C4/C2=C2

G = SD₁₆⋊C₄ order 64 = 2⁶

1^st semidirect product of SD₁₆ and C₄ acting via C₄/C₂=C₂