C3^2:6C4wrC2 - GroupNames

G = C₃²⋊₆C₄≀C₂ order 288 = 2⁵·3²

The semidirect product of C₃² and C₄≀C₂ acting via C₄≀C₂/D₄=C₄

Aliases: C₃²⋊₆C₄≀C₂, D₄⋊₂(C₃²⋊C₄), (D₄×C₃²)⋊₂C₄, C₃²⋊₄Q₈⋊₁C₄, C₃⋊Dic₃.₅₂D₄, C₂.₇(C₆²⋊C₄), C₁₂.D₆.₁C₂, C₃²⋊M₄(2)⋊₂C₂, (C₄×C₃²⋊C₄)⋊₁C₂, C₄.₂(C₂×C₃²⋊C₄), (C₃×C₁₂).₂(C₂×C₄), (C₂×C₃⋊S₃).₁₄D₄, (C₄×C₃⋊S₃).₆C₂², (C₃×C₆).₁₇(C₂²⋊C₄), SmallGroup(288,431)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃²⋊₆C₄≀C₂

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₄×C₃⋊S₃ — C₃²⋊M₄(2) — C₃²⋊₆C₄≀C₂

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃²⋊₆C₄≀C₂

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for C₃²⋊₆C₄≀C₂
G = < a,b,c,d,e | a³=b³=c⁴=d²=e⁴=1, ebe^-1=ab=ba, ac=ca, ad=da, eae^-1=a^-1b, bc=cb, bd=db, dcd=c^-1, ce=ec, ede^-1=c^-1d >

Subgroups: 456 in 84 conjugacy classes, 16 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₈, C₂×C₄, D₄, D₄, Q₈, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₄², M₄(2), C₄○D₄, C₃⋊S₃, C₃×C₆, C₃×C₆, Dic₆, C₄×S₃, C₂×Dic₃, C₃⋊D₄, C₃×D₄, C₄≀C₂, C₃⋊Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃²⋊C₄, C₂×C₃⋊S₃, C₆², D₄⋊₂S₃, C₃²⋊₂C₈, C₃²⋊₄Q₈, C₄×C₃⋊S₃, C₂×C₃⋊Dic₃, C₃²⋊₇D₄, D₄×C₃², C₂×C₃²⋊C₄, C₃²⋊M₄(2), C₄×C₃²⋊C₄, C₁₂.D₆, C₃²⋊₆C₄≀C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, C₄≀C₂, C₃²⋊C₄, C₂×C₃²⋊C₄, C₆²⋊C₄, C₃²⋊₆C₄≀C₂

Character table of C₃²⋊₆C₄≀C₂

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	6E	6F	8A	8B	12A	12B
size	1	1	4	18	4	4	2	9	9	18	18	18	18	36	4	4	8	8	8	8	36	36	8	8
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	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	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	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	linear of order 2
ρ₅	1	1	1	-1	1	1	1	-1	-1	-i	i	-i	i	-1	1	1	1	1	1	1	i	-i	1	1	linear of order 4
ρ₆	1	1	-1	-1	1	1	1	-1	-1	-i	i	-i	i	1	1	1	-1	-1	-1	-1	-i	i	1	1	linear of order 4
ρ₇	1	1	1	-1	1	1	1	-1	-1	i	-i	i	-i	-1	1	1	1	1	1	1	-i	i	1	1	linear of order 4
ρ₈	1	1	-1	-1	1	1	1	-1	-1	i	-i	i	-i	1	1	1	-1	-1	-1	-1	i	-i	1	1	linear of order 4
ρ₉	2	2	0	2	2	2	-2	-2	-2	0	0	0	0	0	2	2	0	0	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	0	-2	2	2	-2	2	2	0	0	0	0	0	2	2	0	0	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	-2	0	0	2	2	0	2i	-2i	-1-i	1-i	1+i	-1+i	0	-2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₂	2	-2	0	0	2	2	0	2i	-2i	1+i	-1+i	-1-i	1-i	0	-2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₃	2	-2	0	0	2	2	0	-2i	2i	1-i	-1-i	-1+i	1+i	0	-2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	-2	0	0	2	2	0	-2i	2i	-1+i	1+i	1-i	-1-i	0	-2	-2	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₅	4	4	0	0	1	-2	-4	0	0	0	0	0	0	0	1	-2	3	0	-3	0	0	0	-1	2	orthogonal lifted from C₆²⋊C₄
ρ₁₆	4	4	-4	0	1	-2	4	0	0	0	0	0	0	0	1	-2	-1	2	-1	2	0	0	1	-2	orthogonal lifted from C₂×C₃²⋊C₄
ρ₁₇	4	4	-4	0	-2	1	4	0	0	0	0	0	0	0	-2	1	2	-1	2	-1	0	0	-2	1	orthogonal lifted from C₂×C₃²⋊C₄
ρ₁₈	4	4	4	0	-2	1	4	0	0	0	0	0	0	0	-2	1	-2	1	-2	1	0	0	-2	1	orthogonal lifted from C₃²⋊C₄
ρ₁₉	4	4	0	0	-2	1	-4	0	0	0	0	0	0	0	-2	1	0	-3	0	3	0	0	2	-1	orthogonal lifted from C₆²⋊C₄
ρ₂₀	4	4	4	0	1	-2	4	0	0	0	0	0	0	0	1	-2	1	-2	1	-2	0	0	1	-2	orthogonal lifted from C₃²⋊C₄
ρ₂₁	4	4	0	0	1	-2	-4	0	0	0	0	0	0	0	1	-2	-3	0	3	0	0	0	-1	2	orthogonal lifted from C₆²⋊C₄
ρ₂₂	4	4	0	0	-2	1	-4	0	0	0	0	0	0	0	-2	1	0	3	0	-3	0	0	2	-1	orthogonal lifted from C₆²⋊C₄
ρ₂₃	8	-8	0	0	2	-4	0	0	0	0	0	0	0	0	-2	4	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₂₄	8	-8	0	0	-4	2	0	0	0	0	0	0	0	0	4	-2	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₃²⋊₆C₄≀C₂
►On 48 points

Generators in S₄₈

(5 10 47)(6 11 48)(7 12 45)(8 9 46)(33 37 42)(34 38 43)(35 39 44)(36 40 41)

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

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

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

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

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

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

Matrix representation of C₃²⋊₆C₄≀C₂ ►in GL₆(𝔽₇₃)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	72	72	0	0
0	0	0	0	72	72
0	0	0	0	1	0

27	0	0	0	0	0
46	46	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	1

27	54	0	0	0	0
46	46	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	1

46	0	0	0	0	0
14	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	72	72	0	0

G:=sub<GL(6,GF(73))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[27,46,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,46,0,0,0,0,54,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₃²⋊₆C₄≀C₂ in GAP, Magma, Sage, TeX

C_3^2\rtimes_6C_4\wr C_2

% in TeX

G:=Group("C3^2:6C4wrC2");

// GroupNames label

G:=SmallGroup(288,431);

// by ID

G=gap.SmallGroup(288,431);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,100,675,346,80,9413,691,12550,2372]);

// Polycyclic

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

// generators/relations

Export

Character table of C₃²⋊₆C₄≀C₂ in TeX

G = C32⋊6C4≀C2 order 288 = 25·32

The semidirect product of C32 and C4≀C2 acting via C4≀C2/D4=C4

G = C₃²⋊₆C₄≀C₂ order 288 = 2⁵·3²

The semidirect product of C₃² and C₄≀C₂ acting via C₄≀C₂/D₄=C₄