C2xC3^2:D8 - GroupNames

G = C₂xC₃²:D₈ order 288 = 2⁵·3²

Direct product of C₂ and C₃²:D₈

direct product, non-abelian, soluble, monomial

Aliases: C₂xC₃²:D₈, C₆².₁₁D₄, (C₃xC₆):D₈, C₃²:₂(C₂xD₈), C₂².₁₄S₃wrC₂, C₃:Dic₃.₂₉D₄, D₆:S₃:₉C₂², C₃²:₂C₈:₄C₂², C₃:Dic₃.₇C₂³, C₂.₁₆(C₂xS₃wrC₂), (C₃xC₆).₁₆(C₂xD₄), (C₂xC₃²:₂C₈):₄C₂, (C₂xD₆:S₃):₁₁C₂, (C₂xC₃:Dic₃).₉₄C₂², SmallGroup(288,883)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃:Dic₃ — C₂xC₃²:D₈

Chief series

C₁ — C₃² — C₃xC₆ — C₃:Dic₃ — D₆:S₃ — C₃²:D₈ — C₂xC₃²:D₈

Lower central

C₃² — C₃xC₆ — C₃:Dic₃ — C₂xC₃²:D₈

Upper central

C₁ — C₂²

Generators and relations for C₂xC₃²:D₈
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, ebe=dcd^-1=b^-1, ce=ec, ede=d^-1 >

Subgroups: 688 in 130 conjugacy classes, 27 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₈, C₂xC₄, D₄, C₂³, C₃², Dic₃, D₆, C₂xC₆, C₂xC₈, D₈, C₂xD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₂xDic₃, C₃:D₄, C₂²xS₃, C₂²xC₆, C₂xD₈, C₃:Dic₃, S₃xC₆, C₆², C₂xC₃:D₄, C₃²:₂C₈, D₆:S₃, D₆:S₃, C₂xC₃:Dic₃, S₃xC₂xC₆, C₃²:D₈, C₂xC₃²:₂C₈, C₂xD₆:S₃, C₂xC₃²:D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂xD₄, C₂xD₈, S₃wrC₂, C₃²:D₈, C₂xS₃wrC₂, C₂xC₃²:D₈

Character table of C₂xC₃²:D₈

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	6N	8A	8B	8C	8D
size	1	1	1	1	12	12	12	12	4	4	18	18	4	4	4	4	4	4	12	12	12	12	12	12	12	12	18	18	18	18
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	0	0	2	2	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	0	0	2	2	-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	0	0	2	2	0	0	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	-2	-2	2	0	0	0	0	2	2	0	0	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₃	2	-2	-2	2	0	0	0	0	2	2	0	0	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₄	2	-2	2	-2	0	0	0	0	2	2	0	0	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₅	4	4	4	4	-2	0	0	-2	-2	1	0	0	-2	1	1	-2	-2	1	0	0	0	1	1	0	1	1	0	0	0	0	orthogonal lifted from S₃wrC₂
ρ₁₆	4	4	4	4	2	0	0	2	-2	1	0	0	-2	1	1	-2	-2	1	0	0	0	-1	-1	0	-1	-1	0	0	0	0	orthogonal lifted from S₃wrC₂
ρ₁₇	4	4	4	4	0	-2	-2	0	1	-2	0	0	1	-2	-2	1	1	-2	1	1	1	0	0	1	0	0	0	0	0	0	orthogonal lifted from S₃wrC₂
ρ₁₈	4	4	-4	-4	0	2	-2	0	1	-2	0	0	-1	2	-2	1	-1	2	-1	1	-1	0	0	1	0	0	0	0	0	0	orthogonal lifted from C₂xS₃wrC₂
ρ₁₉	4	4	-4	-4	2	0	0	-2	-2	1	0	0	2	-1	1	-2	2	-1	0	0	0	1	1	0	-1	-1	0	0	0	0	orthogonal lifted from C₂xS₃wrC₂
ρ₂₀	4	4	-4	-4	-2	0	0	2	-2	1	0	0	2	-1	1	-2	2	-1	0	0	0	-1	-1	0	1	1	0	0	0	0	orthogonal lifted from C₂xS₃wrC₂
ρ₂₁	4	4	-4	-4	0	-2	2	0	1	-2	0	0	-1	2	-2	1	-1	2	1	-1	1	0	0	-1	0	0	0	0	0	0	orthogonal lifted from C₂xS₃wrC₂
ρ₂₂	4	4	4	4	0	2	2	0	1	-2	0	0	1	-2	-2	1	1	-2	-1	-1	-1	0	0	-1	0	0	0	0	0	0	orthogonal lifted from S₃wrC₂
ρ₂₃	4	-4	-4	4	0	0	0	0	-2	1	0	0	-2	-1	-1	2	2	1	0	0	0	√-3	-√-3	0	-√-3	√-3	0	0	0	0	complex lifted from C₃²:D₈
ρ₂₄	4	-4	4	-4	0	0	0	0	-2	1	0	0	2	1	-1	2	-2	-1	0	0	0	√-3	-√-3	0	√-3	-√-3	0	0	0	0	complex lifted from C₃²:D₈
ρ₂₅	4	-4	-4	4	0	0	0	0	1	-2	0	0	1	2	2	-1	-1	-2	√-3	√-3	-√-3	0	0	-√-3	0	0	0	0	0	0	complex lifted from C₃²:D₈
ρ₂₆	4	-4	4	-4	0	0	0	0	1	-2	0	0	-1	-2	2	-1	1	2	√-3	-√-3	-√-3	0	0	√-3	0	0	0	0	0	0	complex lifted from C₃²:D₈
ρ₂₇	4	-4	4	-4	0	0	0	0	1	-2	0	0	-1	-2	2	-1	1	2	-√-3	√-3	√-3	0	0	-√-3	0	0	0	0	0	0	complex lifted from C₃²:D₈
ρ₂₈	4	-4	-4	4	0	0	0	0	1	-2	0	0	1	2	2	-1	-1	-2	-√-3	-√-3	√-3	0	0	√-3	0	0	0	0	0	0	complex lifted from C₃²:D₈
ρ₂₉	4	-4	-4	4	0	0	0	0	-2	1	0	0	-2	-1	-1	2	2	1	0	0	0	-√-3	√-3	0	√-3	-√-3	0	0	0	0	complex lifted from C₃²:D₈
ρ₃₀	4	-4	4	-4	0	0	0	0	-2	1	0	0	2	1	-1	2	-2	-1	0	0	0	-√-3	√-3	0	-√-3	√-3	0	0	0	0	complex lifted from C₃²:D₈

Smallest permutation representation of C₂xC₃²:D₈
►On 48 points

Generators in S₄₈

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

(1 35 22)(3 24 37)(5 39 18)(7 20 33)(9 46 26)(11 28 48)(13 42 30)(15 32 44)

(2 23 36)(4 38 17)(6 19 40)(8 34 21)(10 27 47)(12 41 29)(14 31 43)(16 45 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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)

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

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

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

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

Matrix representation of C₂xC₃²:D₈ ►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	72	1	0	0
0	0	72	0	0	0
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	0	0
0	0	0	1	0	0
0	0	0	0	72	1
0	0	0	0	72	0

33	50	0	0	0	0
22	40	0	0	0	0
0	0	0	0	30	13
0	0	0	0	60	43
0	0	0	1	0	0
0	0	1	0	0	0

1	0	0	0	0	0
60	72	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	30	13
0	0	0	0	60	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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[33,22,0,0,0,0,50,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,30,60,0,0,0,0,13,43,0,0],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;

C₂xC₃²:D₈ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes D_8

% in TeX

G:=Group("C2xC3^2:D8");

// GroupNames label

G:=SmallGroup(288,883);

// by ID

G=gap.SmallGroup(288,883);

# by ID

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

// 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,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;

// generators/relations

Export

Character table of C₂xC₃²:D₈ in TeX

G = C2xC32:D8 order 288 = 25·32

Direct product of C2 and C32:D8

G = C₂xC₃²:D₈ order 288 = 2⁵·3²

Direct product of C₂ and C₃²:D₈