C2^3xC3^2: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

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

Generators and relations
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 >

Smallest permutation representation
►On 48 points

Generators in S₄₈

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

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

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

(1 40 38)(4 43 41)(6 27 25)(8 46 48)(10 29 31)(11 21 23)(14 18 20)(15 34 36)

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

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

Matrix representation ►G ⊆ 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] >;

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

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

G = C₂³×C₃²⋊C₄ order 288 = 2⁵·3²

Direct product of C₂³ and C₃²⋊C₄