C2^3xC3^2:C4 - GroupNames

G = C₂³×C₃²⋊C₄ order 288 = 2⁵·3²

Direct product of C₂³ and C₃²⋊C₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂³×C₃²⋊C₄, (C₂×C₆²)⋊₆C₄, C₆²⋊₃(C₂×C₄), C₃⋊S₃.₃C₂⁴, C₃²⋊₂(C₂³×C₄), C₃⋊S₃⋊₃(C₂²×C₄), (C₃×C₆)⋊₂(C₂²×C₄), (C₂²×C₃⋊S₃)⋊₁₀C₄, (C₂³×C₃⋊S₃).₇C₂, (C₂×C₃⋊S₃).₅₈C₂³, (C₂²×C₃⋊S₃).₁₁₀C₂², (C₂×C₃⋊S₃)⋊₂₀(C₂×C₄), SmallGroup(288,1039)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂³×C₃²⋊C₄

Chief series

C₁ — C₃² — C₃⋊S₃ — C₃²⋊C₄ — C₂×C₃²⋊C₄ — C₂²×C₃²⋊C₄ — C₂³×C₃²⋊C₄

Lower central

C₃² — C₂³×C₃²⋊C₄

Upper central

C₁ — C₂³

Generators and relations for C₂³×C₃²⋊C₄
G = < a,b,c,d,e,f | a²=b²=c²=d³=e³=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef^-1=de=ed, fdf^-1=d^-1e >

Subgroups: 1728 in 370 conjugacy classes, 134 normal (7 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₂×C₄, C₂³, C₂³, C₃², D₆, C₂×C₆, C₂²×C₄, C₂⁴, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₂²×S₃, C₂²×C₆, C₂³×C₄, C₃²⋊C₄, C₂×C₃⋊S₃, C₆², S₃×C₂³, C₂×C₃²⋊C₄, C₂²×C₃⋊S₃, C₂×C₆², C₂²×C₃²⋊C₄, C₂³×C₃⋊S₃, C₂³×C₃²⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, C₂³×C₄, C₃²⋊C₄, C₂×C₃²⋊C₄, C₂²×C₃²⋊C₄, C₂³×C₃²⋊C₄

Smallest permutation representation of C₂³×C₃²⋊C₄
►On 48 points

Generators in S₄₈

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

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

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

(1 28 26)(3 44 42)(6 30 32)(8 35 33)(9 47 45)(11 17 19)(14 23 21)(16 40 38)

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

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

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

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

48 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3A	3B	4A	···	4P	6A	···	6N
order	1	2	···	2	2	···	2	3	3	4	···	4	6	···	6
size	1	1	···	1	9	···	9	4	4	9	···	9	4	···	4

48 irreducible representations

dim	1	1	1	1	1	4	4
type	+	+	+			+	+
image	C₁	C₂	C₂	C₄	C₄	C₃²⋊C₄	C₂×C₃²⋊C₄
kernel	C₂³×C₃²⋊C₄	C₂²×C₃²⋊C₄	C₂³×C₃⋊S₃	C₂²×C₃⋊S₃	C₂×C₆²	C₂³	C₂²
# reps	1	14	1	14	2	2	14

Matrix representation of C₂³×C₃²⋊C₄ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

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	12	1
0	0	0	0	12	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	12	0	0
0	0	1	12	0	0
0	0	0	0	12	1
0	0	0	0	12	0

5	0	0	0	0	0
0	8	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³×C₃²⋊C₄ in GAP, Magma, Sage, TeX

C_2^3\times C_3^2\rtimes C_4

% in TeX

G:=Group("C2^3xC3^2:C4");

// GroupNames label

G:=SmallGroup(288,1039);

// by ID

G=gap.SmallGroup(288,1039);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,9413,201,12550,622]);

// Polycyclic

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

// generators/relations

G = C23×C32⋊C4 order 288 = 25·32

Direct product of C23 and C32⋊C4

G = C₂³×C₃²⋊C₄ order 288 = 2⁵·3²

Direct product of C₂³ and C₃²⋊C₄