C2xC3^2:2SD16 - GroupNames

G = C₂×C₃²⋊₂SD₁₆ order 288 = 2⁵·3²

Direct product of C₂ and C₃²⋊₂SD₁₆

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₃²⋊₂SD₁₆, C₆².₁₄D₄, (C₃×C₆)⋊₂SD₁₆, C₃²⋊₃(C₂×SD₁₆), C₂².₁₅S₃≀C₂, C₃⋊Dic₃.₃₂D₄, C₃²⋊₂C₈⋊₆C₂², C₃²⋊₂Q₈⋊₈C₂², D₆⋊S₃.₇C₂², C₃⋊Dic₃.₁₀C₂³, C₂.₁₉(C₂×S₃≀C₂), (C₃×C₆).₁₉(C₂×D₄), (C₂×C₃²⋊₂C₈)⋊₆C₂, (C₂×C₃²⋊₂Q₈)⋊₉C₂, (C₂×D₆⋊S₃).₈C₂, (C₂×C₃⋊Dic₃).₉₇C₂², SmallGroup(288,886)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊Dic₃ — C₂×C₃²⋊₂SD₁₆

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — D₆⋊S₃ — C₃²⋊₂SD₁₆ — C₂×C₃²⋊₂SD₁₆

Lower central

C₃² — C₃×C₆ — C₃⋊Dic₃ — C₂×C₃²⋊₂SD₁₆

Upper central

C₁ — C₂²

Generators and relations for C₂×C₃²⋊₂SD₁₆
G = < a,b,c,d,e | a²=b³=c³=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=c^-1, ebe=b^-1, dcd^-1=b, ce=ec, ede=d³ >

Subgroups: 560 in 114 conjugacy classes, 27 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₈, C₂×C₄, D₄, Q₈, C₂³, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, C₃×S₃, C₃×C₆, C₃×C₆, Dic₆, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂²×S₃, C₂²×C₆, C₂×SD₁₆, C₃×Dic₃, C₃⋊Dic₃, S₃×C₆, C₆², C₂×Dic₆, C₂×C₃⋊D₄, C₃²⋊₂C₈, D₆⋊S₃, D₆⋊S₃, C₃²⋊₂Q₈, C₃²⋊₂Q₈, C₆×Dic₃, C₂×C₃⋊Dic₃, S₃×C₂×C₆, C₃²⋊₂SD₁₆, C₂×C₃²⋊₂C₈, C₂×D₆⋊S₃, C₂×C₃²⋊₂Q₈, C₂×C₃²⋊₂SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₂×SD₁₆, S₃≀C₂, C₃²⋊₂SD₁₆, C₂×S₃≀C₂, C₂×C₃²⋊₂SD₁₆

Character table of C₂×C₃²⋊₂SD₁₆

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	8A	8B	8C	8D	12A	12B	12C	12D
size	1	1	1	1	12	12	4	4	12	12	18	18	4	4	4	4	4	4	12	12	12	12	18	18	18	18	12	12	12	12
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	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	-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	-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	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	-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	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	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	-1	-1	linear of order 2
ρ₉	2	2	-2	-2	0	0	2	2	0	0	2	-2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	2	0	0	-2	-2	2	2	2	2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	0	2	2	0	0	0	0	2	-2	-2	-2	-2	2	0	0	0	0	-√-2	√-2	√-2	-√-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₂	2	-2	2	-2	0	0	2	2	0	0	0	0	-2	2	2	-2	-2	-2	0	0	0	0	-√-2	-√-2	√-2	√-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₃	2	-2	-2	2	0	0	2	2	0	0	0	0	2	-2	-2	-2	-2	2	0	0	0	0	√-2	-√-2	-√-2	√-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₄	2	-2	2	-2	0	0	2	2	0	0	0	0	-2	2	2	-2	-2	-2	0	0	0	0	√-2	√-2	-√-2	-√-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₅	4	4	4	4	2	2	1	-2	0	0	0	0	1	1	-2	1	-2	-2	-1	-1	-1	-1	0	0	0	0	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₆	4	4	4	4	0	0	-2	1	2	2	0	0	-2	-2	1	-2	1	1	0	0	0	0	0	0	0	0	-1	-1	-1	-1	orthogonal lifted from S₃≀C₂
ρ₁₇	4	4	4	4	0	0	-2	1	-2	-2	0	0	-2	-2	1	-2	1	1	0	0	0	0	0	0	0	0	1	1	1	1	orthogonal lifted from S₃≀C₂
ρ₁₈	4	4	4	4	-2	-2	1	-2	0	0	0	0	1	1	-2	1	-2	-2	1	1	1	1	0	0	0	0	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₉	4	4	-4	-4	-2	2	1	-2	0	0	0	0	-1	-1	2	1	-2	2	-1	-1	1	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×S₃≀C₂
ρ₂₀	4	4	-4	-4	0	0	-2	1	-2	2	0	0	2	2	-1	-2	1	-1	0	0	0	0	0	0	0	0	-1	-1	1	1	orthogonal lifted from C₂×S₃≀C₂
ρ₂₁	4	4	-4	-4	0	0	-2	1	2	-2	0	0	2	2	-1	-2	1	-1	0	0	0	0	0	0	0	0	1	1	-1	-1	orthogonal lifted from C₂×S₃≀C₂
ρ₂₂	4	4	-4	-4	2	-2	1	-2	0	0	0	0	-1	-1	2	1	-2	2	1	1	-1	-1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×S₃≀C₂
ρ₂₃	4	-4	4	-4	0	0	-2	1	0	0	0	0	2	-2	1	2	-1	-1	0	0	0	0	0	0	0	0	√3	-√3	-√3	√3	symplectic lifted from C₃²⋊₂SD₁₆, Schur index 2
ρ₂₄	4	-4	-4	4	0	0	-2	1	0	0	0	0	-2	2	-1	2	-1	1	0	0	0	0	0	0	0	0	-√3	√3	-√3	√3	symplectic lifted from C₃²⋊₂SD₁₆, Schur index 2
ρ₂₅	4	-4	4	-4	0	0	-2	1	0	0	0	0	2	-2	1	2	-1	-1	0	0	0	0	0	0	0	0	-√3	√3	√3	-√3	symplectic lifted from C₃²⋊₂SD₁₆, Schur index 2
ρ₂₆	4	-4	-4	4	0	0	-2	1	0	0	0	0	-2	2	-1	2	-1	1	0	0	0	0	0	0	0	0	√3	-√3	√3	-√3	symplectic lifted from C₃²⋊₂SD₁₆, Schur index 2
ρ₂₇	4	-4	4	-4	0	0	1	-2	0	0	0	0	-1	1	-2	-1	2	2	-√-3	√-3	√-3	-√-3	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂SD₁₆
ρ₂₈	4	-4	-4	4	0	0	1	-2	0	0	0	0	1	-1	2	-1	2	-2	-√-3	√-3	-√-3	√-3	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂SD₁₆
ρ₂₉	4	-4	4	-4	0	0	1	-2	0	0	0	0	-1	1	-2	-1	2	2	√-3	-√-3	-√-3	√-3	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂SD₁₆
ρ₃₀	4	-4	-4	4	0	0	1	-2	0	0	0	0	1	-1	2	-1	2	-2	√-3	-√-3	√-3	-√-3	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂SD₁₆

Smallest permutation representation of C₂×C₃²⋊₂SD₁₆
►On 48 points

Generators in S₄₈

(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)

(1 45 31)(3 25 47)(5 41 27)(7 29 43)(9 18 35)(11 37 20)(13 22 39)(15 33 24)

(2 46 32)(4 26 48)(6 42 28)(8 30 44)(10 19 36)(12 38 21)(14 23 40)(16 34 17)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)

(1 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 41)(10 44)(11 47)(12 42)(13 45)(14 48)(15 43)(16 46)(25 37)(26 40)(27 35)(28 38)(29 33)(30 36)(31 39)(32 34)

G:=sub<Sym(48)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (1,45,31)(3,25,47)(5,41,27)(7,29,43)(9,18,35)(11,37,20)(13,22,39)(15,33,24), (2,46,32)(4,26,48)(6,42,28)(8,30,44)(10,19,36)(12,38,21)(14,23,40)(16,34,17), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,22)(2,17)(3,20)(4,23)(5,18)(6,21)(7,24)(8,19)(9,41)(10,44)(11,47)(12,42)(13,45)(14,48)(15,43)(16,46)(25,37)(26,40)(27,35)(28,38)(29,33)(30,36)(31,39)(32,34)>;

G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (1,45,31)(3,25,47)(5,41,27)(7,29,43)(9,18,35)(11,37,20)(13,22,39)(15,33,24), (2,46,32)(4,26,48)(6,42,28)(8,30,44)(10,19,36)(12,38,21)(14,23,40)(16,34,17), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,22)(2,17)(3,20)(4,23)(5,18)(6,21)(7,24)(8,19)(9,41)(10,44)(11,47)(12,42)(13,45)(14,48)(15,43)(16,46)(25,37)(26,40)(27,35)(28,38)(29,33)(30,36)(31,39)(32,34) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)], [(1,45,31),(3,25,47),(5,41,27),(7,29,43),(9,18,35),(11,37,20),(13,22,39),(15,33,24)], [(2,46,32),(4,26,48),(6,42,28),(8,30,44),(10,19,36),(12,38,21),(14,23,40),(16,34,17)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,22),(2,17),(3,20),(4,23),(5,18),(6,21),(7,24),(8,19),(9,41),(10,44),(11,47),(12,42),(13,45),(14,48),(15,43),(16,46),(25,37),(26,40),(27,35),(28,38),(29,33),(30,36),(31,39),(32,34)]])

Matrix representation of C₂×C₃²⋊₂SD₁₆ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	5	40	1	0
0	0	5	40	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	23	38	72	72
0	0	0	0	1	0

60	50	0	0	0	0
55	13	0	0	0	0
0	0	23	38	71	72
0	0	0	0	72	1
0	0	49	5	45	5
0	0	6	18	45	5

72	0	0	0	0	0
17	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	4	60	30	60
0	0	70	4	13	43

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,5,5,0,0,72,72,40,40,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,0,0,0,0,1,38,0,0,0,0,0,72,1,0,0,0,0,72,0],[60,55,0,0,0,0,50,13,0,0,0,0,0,0,23,0,49,6,0,0,38,0,5,18,0,0,71,72,45,45,0,0,72,1,5,5],[72,17,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4,70,0,0,1,0,60,4,0,0,0,0,30,13,0,0,0,0,60,43] >;

C₂×C₃²⋊₂SD₁₆ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2{\rm SD}_{16}

% in TeX

G:=Group("C2xC3^2:2SD16");

// GroupNames label

G:=SmallGroup(288,886);

// by ID

G=gap.SmallGroup(288,886);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,675,346,80,2693,2028,362,797,1203]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂×C₃²⋊₂SD₁₆ in TeX

G = C2×C32⋊2SD16 order 288 = 25·32

Direct product of C2 and C32⋊2SD16

G = C₂×C₃²⋊₂SD₁₆ order 288 = 2⁵·3²

Direct product of C₂ and C₃²⋊₂SD₁₆