C4xSD16 - GroupNames

G = C₄xSD₁₆ order 64 = 2⁶

Direct product of C₄ and SD₁₆

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

Aliases: C₄xSD₁₆, C₄².₇₂C₂², C₈:₅(C₂xC₄), (C₄xC₈):₁₁C₂, Q₈:₁(C₂xC₄), (C₄xQ₈):₁C₂, C₄ o₂(C₄.Q₈), C₄.Q₈:₁₄C₂, D₄.₁(C₂xC₄), (C₄xD₄).₄C₂, C₂.₁₃(C₄xD₄), (C₂xC₄).₅₂D₄, C₂.₄(C₄oD₈), C₄.₂(C₄oD₄), C₄ o₃(D₄:C₄), C₂.₄(C₂xSD₁₆), C₄ o₂(Q₈:C₄), Q₈:C₄:₂₁C₂, D₄:C₄.₈C₂, C₄:C₄.₅₁C₂², (C₂xC₈).₆₃C₂², C₄.₁₀(C₂²xC₄), (C₂xC₄).₇₄C₂³, (C₂xSD₁₆).₅C₂, C₂².₅₂(C₂xD₄), (C₂xD₄).₅₁C₂², (C₂xQ₈).₄₄C₂², SmallGroup(64,119)

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

Derived series

C₁ — C₄ — C₄xSD₁₆

Chief series

C₁ — C₂ — C₂² — C₂xC₄ — C₄² — C₄xQ₈ — C₄xSD₁₆

Lower central

C₁ — C₂ — C₄ — C₄xSD₁₆

Upper central

C₁ — C₂xC₄ — C₄² — C₄xSD₁₆

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₄xSD₁₆

Generators and relations for C₄xSD₁₆
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, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₂²:C₄, C₄:C₄, C₄:C₄, C₂xC₈, SD₁₆, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄xC₈, D₄:C₄, Q₈:C₄, C₄.Q₈, C₄xD₄, C₄xQ₈, C₂xSD₁₆, C₄xSD₁₆
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, SD₁₆, C₂²xC₄, C₂xD₄, C₄oD₄, C₄xD₄, C₂xSD₁₆, C₄oD₈, C₄xSD₁₆

Character table of C₄xSD₁₆

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 C₄oD₄
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 C₄oD₄
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 C₄oD₈
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 C₄oD₈
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 C₄oD₈
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 C₄oD₈

Smallest permutation representation of C₄xSD₁₆
►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)]])

C₄xSD₁₆ 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.(C₄xD₄): 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 ...
C₄xSD₁₆ is a maximal quotient of
C₈:₁₂SD₁₆ C₈:₁₅SD₁₆ C₈:₉SD₁₆ C₄.Q₈:₉C₄ C₄.Q₈:₁₀C₄ C₈:₇(C₄:C₄)
C_2p.(C₄xD₄): D₄:(C₄:C₄) Q₈:C₄:C₄ (C₂xSD₁₆):₁₄C₄ (C₂xSD₁₆):₁₅C₄ C₄.₆₇(C₄xD₄) C₄.₆₈(C₄xD₄) C₂.(C₈:₈D₄) C₂.(C₈:₇D₄) ...

Matrix representation of C₄xSD₁₆ ►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] >;

C₄xSD₁₆ 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 C₄xSD₁₆ in TeX

G = C4xSD16 order 64 = 26

Direct product of C4 and SD16

G = C₄xSD₁₆ order 64 = 2⁶

Direct product of C₄ and SD₁₆