C4xSD16 - GroupNames

G = C₄×SD₁₆ order 64 = 2⁶

Direct product of C₄ and SD₁₆

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

Aliases: C₄×SD₁₆, C₄².₇₂C₂², C₈⋊₅(C₂×C₄), (C₄×C₈)⋊₁₁C₂, Q₈⋊₁(C₂×C₄), (C₄×Q₈)⋊₁C₂, C₄ ○₂(C₄.Q₈), C₄.Q₈⋊₁₄C₂, D₄.₁(C₂×C₄), (C₄×D₄).₄C₂, C₂.₁₃(C₄×D₄), (C₂×C₄).₅₂D₄, C₂.₄(C₄○D₈), C₄.₂(C₄○D₄), C₄ ○₃(D₄⋊C₄), C₂.₄(C₂×SD₁₆), C₄ ○₂(Q₈⋊C₄), Q₈⋊C₄⋊₂₁C₂, D₄⋊C₄.₈C₂, C₄⋊C₄.₅₁C₂², (C₂×C₈).₆₃C₂², C₄.₁₀(C₂²×C₄), (C₂×C₄).₇₄C₂³, (C₂×SD₁₆).₅C₂, C₂².₅₂(C₂×D₄), (C₂×D₄).₅₁C₂², (C₂×Q₈).₄₄C₂², SmallGroup(64,119)

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

Derived series

C₁ — C₄ — C₄×SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — C₄×SD₁₆

Lower central

C₁ — C₂ — C₄ — C₄×SD₁₆

Upper central

C₁ — C₂×C₄ — 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, ab=ba, ac=ca, cbc=b³ >

Subgroups: 101 in 61 conjugacy classes, 37 normal (25 characteristic)
C₁, C₂, C₂, C₄, 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, C₄×C₈, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₄×D₄, C₄×Q₈, C₂×SD₁₆, C₄×SD₁₆
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, SD₁₆, C₂²×C₄, C₂×D₄, C₄○D₄, C₄×D₄, C₂×SD₁₆, C₄○D₈, C₄×SD₁₆

Character table of C₄×SD₁₆

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	4	4	1	1	1	1	2	2	2	2	4	4	4	4	4	4	2	2	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	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	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	-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	-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	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	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	-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	-1	1	-1	1	-1	1	linear of order 2
ρ₉	1	-1	-1	1	-1	1	-i	i	i	-i	1	i	-i	-1	-1	-i	1	i	-i	i	-i	1	-i	1	i	-1	i	-1	linear of order 4
ρ₁₀	1	-1	-1	1	1	-1	i	-i	-i	i	1	-i	i	-1	1	-i	-1	i	-i	i	i	1	i	1	-i	-1	-i	-1	linear of order 4
ρ₁₁	1	-1	-1	1	1	-1	i	-i	-i	i	1	-i	i	-1	-1	-i	1	-i	i	i	-i	-1	-i	-1	i	1	i	1	linear of order 4
ρ₁₂	1	-1	-1	1	-1	1	-i	i	i	-i	1	i	-i	-1	1	-i	-1	-i	i	i	i	-1	i	-1	-i	1	-i	1	linear of order 4
ρ₁₃	1	-1	-1	1	1	-1	-i	i	i	-i	1	i	-i	-1	-1	i	1	i	-i	-i	i	-1	i	-1	-i	1	-i	1	linear of order 4
ρ₁₄	1	-1	-1	1	-1	1	i	-i	-i	i	1	-i	i	-1	1	i	-1	i	-i	-i	-i	-1	-i	-1	i	1	i	1	linear of order 4
ρ₁₅	1	-1	-1	1	-1	1	i	-i	-i	i	1	-i	i	-1	-1	i	1	-i	i	-i	i	1	i	1	-i	-1	-i	-1	linear of order 4
ρ₁₆	1	-1	-1	1	1	-1	-i	i	i	-i	1	i	-i	-1	1	i	-1	-i	i	-i	-i	1	-i	1	i	-1	i	-1	linear of order 4
ρ₁₇	2	2	2	2	0	0	2	2	2	2	-2	-2	-2	-2	0	0	0	0	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	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	-2	2	0	0	2i	-2i	-2i	2i	-2	2i	-2i	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	0	0	-2i	2i	2i	-2i	-2	-2i	2i	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	2	-2	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₂₂	2	2	-2	-2	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	0	-√2	-√-2	√2	√-2	√2	√-2	-√2	-√-2	complex lifted from C₄○D₈
ρ₂₃	2	-2	2	-2	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₄	2	2	-2	-2	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	0	-√2	√-2	√2	-√-2	√2	-√-2	-√2	√-2	complex lifted from C₄○D₈
ρ₂₅	2	-2	2	-2	0	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₂₆	2	-2	2	-2	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₇	2	2	-2	-2	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	0	√2	-√-2	-√2	√-2	-√2	√-2	√2	-√-2	complex lifted from C₄○D₈
ρ₂₈	2	2	-2	-2	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	0	√2	√-2	-√2	-√-2	-√2	-√-2	√2	√-2	complex lifted from C₄○D₈

Smallest permutation representation of C₄×SD₁₆
►On 32 points

Generators in S₃₂

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

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

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

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

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

C₄×SD₁₆ is a maximal subgroup of
SD₁₆⋊C₈ C₈⋊₁₂SD₁₆ C₈⋊₁₅SD₁₆ D₄.M₄(2) Q₈⋊₂M₄(2) C₈⋊₉SD₁₆ C₈⋊M₄(2) C₄².₂₂₂D₄ C₄².₃₈₄D₄ C₄².₂₂₃D₄ C₄².₂₂₅D₄ C₄².₄₅₀D₄ C₄².₄₅₁D₄ C₄².₂₂₆D₄ C₄².₃₅₂C₂³ C₄².₃₅₃C₂³ C₄².₃₅₄C₂³ C₄².₃₅₅C₂³ C₄².₃₅₇C₂³ C₄².₃₅₉C₂³ C₄².₃₆₀C₂³ C₄².₃₀₈D₄ C₄².₂₅₆D₄ C₄².₃₈₅C₂³ C₄².₃₉₀C₂³ SD₁₆⋊₃D₄ SD₁₆⋊₁₀D₄ D₄⋊₇SD₁₆ C₄².₄₆₁C₂³ D₄⋊₈SD₁₆ C₄².₄₆₇C₂³ C₄².₄₇₂C₂³ C₄².₄₇₃C₂³ C₄².₄₇₈C₂³ C₄².₄₈₀C₂³ D₄⋊₉SD₁₆ C₄².₄₈₆C₂³ C₄².₄₈₉C₂³ C₄².₄₉₂C₂³ C₄².₄₉₄C₂³ C₄².₄₉₈C₂³ Q₈⋊₇SD₁₆ C₄².₅₀₁C₂³ C₄².₅₀₂C₂³ Q₈⋊₈SD₁₆ C₄².₅₀₅C₂³ C₄².₅₀₆C₂³ C₄².₅₀₉C₂³ C₄².₅₁₀C₂³ C₄².₅₁₂C₂³ C₄².₅₁₃C₂³ C₄².₅₁₄C₂³ C₄².₅₁₇C₂³ SD₁₆⋊₄Q₈ SD₁₆⋊Q₈ SD₁₆⋊₂Q₈ SD₁₆⋊₃Q₈ C₄².₇₃C₂³ C₄².₅₃₁C₂³
C_2p.(C₄×D₄): C₄².₂₇₅C₂³ C₄².₂₇₆C₂³ C₄².₂₇₈C₂³ C₄².₂₈₁C₂³ Dic₃⋊₆SD₁₆ Dic₃⋊₇SD₁₆ Dic₃⋊₈SD₁₆ Dic₅⋊₆SD₁₆ ...
C₈⋊_pD₄⋊C₂: C₄².₃₆₅D₄ C₄².₂₅₅D₄ C₄².₃₈₆C₂³ C₄².₃₉₁C₂³ SD₁₆⋊₁D₄ SD₁₆⋊₂D₄ SD₁₆⋊₁₁D₄ C₄².₄₆₆C₂³ ...
C₄×SD₁₆ is a maximal quotient of
C₈⋊₁₂SD₁₆ C₈⋊₁₅SD₁₆ C₈⋊₉SD₁₆ C₄.Q₈⋊₉C₄ C₄.Q₈⋊₁₀C₄ C₈⋊₇(C₄⋊C₄)
C_2p.(C₄×D₄): D₄⋊(C₄⋊C₄) Q₈⋊C₄⋊C₄ (C₂×SD₁₆)⋊₁₄C₄ (C₂×SD₁₆)⋊₁₅C₄ C₄.₆₇(C₄×D₄) C₄.₆₈(C₄×D₄) C₂.(C₈⋊₈D₄) C₂.(C₈⋊₇D₄) ...

Matrix representation of C₄×SD₁₆ ►in GL₃(𝔽₁₇) generated by

13	0	0
0	16	0
0	0	16

1	0	0
0	10	7
0	5	0

1	0	0
0	1	0
0	1	16

G:=sub<GL(3,GF(17))| [13,0,0,0,16,0,0,0,16],[1,0,0,0,10,5,0,7,0],[1,0,0,0,1,1,0,0,16] >;

C₄×SD₁₆ in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("C4xSD16");

// GroupNames label

G:=SmallGroup(64,119);

// by ID

G=gap.SmallGroup(64,119);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,86,963,489,117]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄×SD₁₆ in TeX

G = C4×SD16 order 64 = 26

Direct product of C4 and SD16

G = C₄×SD₁₆ order 64 = 2⁶

Direct product of C₄ and SD₁₆