C3^2:7C4wrC2 - GroupNames

G = C₃²⋊₇C₄≀C₂ order 288 = 2⁵·3²

The semidirect product of C₃² and C₄≀C₂ acting via C₄≀C₂/Q₈=C₄

Aliases: C₃²⋊₇C₄≀C₂, C₁₂⋊S₃⋊₂C₄, Q₈⋊₂(C₃²⋊C₄), (Q₈×C₃²)⋊₂C₄, C₃⋊Dic₃.₅₃D₄, C₂.₉(C₆²⋊C₄), C₁₂.₂₆D₆.₁C₂, C₃²⋊M₄(2)⋊₃C₂, (C₄×C₃²⋊C₄)⋊₂C₂, C₄.₄(C₂×C₃²⋊C₄), (C₃×C₁₂).₄(C₂×C₄), (C₂×C₃⋊S₃).₁₅D₄, (C₄×C₃⋊S₃).₈C₂², (C₃×C₆).₁₉(C₂²⋊C₄), SmallGroup(288,433)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃²⋊₇C₄≀C₂

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₄×C₃⋊S₃ — C₃²⋊M₄(2) — C₃²⋊₇C₄≀C₂

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃²⋊₇C₄≀C₂

Upper central

C₁ — C₂ — C₄ — Q₈

Generators and relations for C₃²⋊₇C₄≀C₂
G = < a,b,c,d,e | a³=b³=c⁴=d²=e⁴=1, ab=ba, ac=ca, dad=a^-1, eae^-1=ab^-1, bc=cb, dbd=b^-1, ebe^-1=a^-1b^-1, dcd=c^-1, ce=ec, ede^-1=c^-1d >

Subgroups: 520 in 84 conjugacy classes, 16 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₈, C₂×C₄, D₄, Q₈, C₃², Dic₃, C₁₂, D₆, C₄², M₄(2), C₄○D₄, C₃⋊S₃, C₃×C₆, C₄×S₃, D₁₂, C₃×Q₈, C₄≀C₂, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, C₃²⋊C₄, C₂×C₃⋊S₃, C₂×C₃⋊S₃, Q₈⋊₃S₃, C₃²⋊₂C₈, C₄×C₃⋊S₃, C₄×C₃⋊S₃, C₁₂⋊S₃, C₁₂⋊S₃, Q₈×C₃², C₂×C₃²⋊C₄, C₃²⋊M₄(2), C₄×C₃²⋊C₄, C₁₂.₂₆D₆, C₃²⋊₇C₄≀C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, C₄≀C₂, C₃²⋊C₄, C₂×C₃²⋊C₄, C₆²⋊C₄, C₃²⋊₇C₄≀C₂

Character table of C₃²⋊₇C₄≀C₂

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

Smallest permutation representation of C₃²⋊₇C₄≀C₂
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

Matrix representation of C₃²⋊₇C₄≀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	0	72
0	0	0	0	1	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	0	72
0	0	0	0	1	72

0	1	0	0	0	0
72	0	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

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

13	60	0	0	0	0
13	13	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	72	0	0	0
0	0	0	72	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,13,0,0,0,0,60,13,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,1,0,0,0] >;

C₃²⋊₇C₄≀C₂ in GAP, Magma, Sage, TeX

C_3^2\rtimes_7C_4\wr C_2

% in TeX

G:=Group("C3^2:7C4wrC2");

// GroupNames label

G:=SmallGroup(288,433);

// by ID

G=gap.SmallGroup(288,433);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,100,675,346,80,9413,691,12550,2372]);

// Polycyclic

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

// generators/relations

Export

Character table of C₃²⋊₇C₄≀C₂ in TeX

G = C32⋊7C4≀C2 order 288 = 25·32

The semidirect product of C32 and C4≀C2 acting via C4≀C2/Q8=C4

G = C₃²⋊₇C₄≀C₂ order 288 = 2⁵·3²

The semidirect product of C₃² and C₄≀C₂ acting via C₄≀C₂/Q₈=C₄