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^[×7], C₂^[×8], C₃^[×2], C₄^[×8], C₂²^[×7], C₂²^[×28], S₃^[×16], C₆^[×14], C₂×C₄^[×28], C₂³, C₂³^[×14], C₃², D₆^[×56], C₂×C₆^[×14], C₂²×C₄^[×14], C₂⁴, C₃⋊S₃, C₃⋊S₃^[×7], C₃×C₆^[×7], C₂²×S₃^[×28], C₂²×C₆^[×2], C₂³×C₄, C₃²⋊C₄^[×8], C₂×C₃⋊S₃^[×28], C₆²^[×7], S₃×C₂³^[×2], C₂×C₃²⋊C₄^[×28], C₂²×C₃⋊S₃^[×14], C₂×C₆², C₂²×C₃²⋊C₄^[×14], C₂³×C₃⋊S₃, C₂³×C₃²⋊C₄
Quotients: C₁, C₂^[×15], C₄^[×8], C₂²^[×35], C₂×C₄^[×28], C₂³^[×15], C₂²×C₄^[×14], C₂⁴, C₂³×C₄, C₃²⋊C₄, C₂×C₃²⋊C₄^[×7], C₂²×C₃²⋊C₄^[×7], C₂³×C₃²⋊C₄

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

Generators in S₄₈

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

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

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

(2 48 46)(4 31 29)(5 36 34)(8 23 21)(9 37 39)(11 17 19)(13 28 26)(15 42 44)

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

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