C3xC2^3:2D6 - GroupNames

G = C₃×C₂³⋊₂D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³⋊₂D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₂³⋊₂D₆, C₆²⋊₁₀D₄, C₆².₂₀₂C₂³, (C₆×D₄)⋊₈C₆, D₆⋊₅(C₃×D₄), D₆⋊C₄⋊₁₄C₆, (C₆×D₄)⋊₁₀S₃, (S₃×C₆)⋊₁₈D₄, (C₂×C₁₂)⋊₁₅D₆, C₂³⋊₂(S₃×C₆), C₆.₄₉(C₆×D₄), (C₂²×C₆)⋊₄D₆, (S₃×C₂³)⋊₅C₆, C₆.₁₉₈(S₃×D₄), (C₆×C₁₂)⋊₂₄C₂², C₃²⋊₁₀C₂²≀C₂, (C₂×C₆²)⋊₄C₂², C₆.D₄⋊₁₀C₆, (C₆×Dic₃)⋊₁₈C₂², (C₂×C₄)⋊₂(S₃×C₆), (D₄×C₃×C₆)⋊₁₅C₂, (C₂×C₆)⋊₄(C₃×D₄), C₂.₂₅(C₃×S₃×D₄), (C₂×C₁₂)⋊₇(C₂×C₆), (C₂×D₄)⋊₃(C₃×S₃), (C₂×C₃⋊D₄)⋊₄C₆, (S₃×C₂²×C₆)⋊₄C₂, (C₃×D₆⋊C₄)⋊₃₅C₂, C₃⋊₂(C₃×C₂²≀C₂), (C₆×C₃⋊D₄)⋊₁₈C₂, (C₂²×C₆)⋊₄(C₂×C₆), C₂.₁₃(C₆×C₃⋊D₄), C₂²⋊₄(C₃×C₃⋊D₄), C₂².₅₉(S₃×C₂×C₆), (C₂×C₆)⋊₁₃(C₃⋊D₄), (C₂×Dic₃)⋊₂(C₂×C₆), (C₃×C₆).₂₅₉(C₂×D₄), C₆.₁₅₀(C₂×C₃⋊D₄), (S₃×C₂×C₆).₉₉C₂², (C₂×C₆).₅₇(C₂²×C₆), (C₃×C₆.D₄)⋊₂₆C₂, (C₂²×S₃).₂₆(C₂×C₆), (C₂×C₆).₃₃₅(C₂²×S₃), SmallGroup(288,708)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₃×C₂³⋊₂D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₆² — S₃×C₂×C₆ — S₃×C₂²×C₆ — C₃×C₂³⋊₂D₆

Lower central

C₃ — C₂×C₆ — C₃×C₂³⋊₂D₆

Upper central

C₁ — C₂×C₆ — C₆×D₄

Generators and relations for C₃×C₂³⋊₂D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁶=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 778 in 287 conjugacy classes, 74 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₂⁴, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²≀C₂, C₃×Dic₃, C₃×C₁₂, S₃×C₆, S₃×C₆, C₆², C₆², C₆², D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₂×C₃⋊D₄, C₆×D₄, C₆×D₄, S₃×C₂³, C₂³×C₆, C₆×Dic₃, C₃×C₃⋊D₄, C₆×C₁₂, D₄×C₃², S₃×C₂×C₆, S₃×C₂×C₆, C₂×C₆², C₂³⋊₂D₆, C₃×C₂²≀C₂, C₃×D₆⋊C₄, C₃×C₆.D₄, C₆×C₃⋊D₄, D₄×C₃×C₆, S₃×C₂²×C₆, C₃×C₂³⋊₂D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂×C₆, C₂×D₄, C₃×S₃, C₃⋊D₄, C₃×D₄, C₂²×S₃, C₂²×C₆, C₂²≀C₂, S₃×C₆, S₃×D₄, C₂×C₃⋊D₄, C₆×D₄, C₃×C₃⋊D₄, S₃×C₂×C₆, C₂³⋊₂D₆, C₃×C₂²≀C₂, C₃×S₃×D₄, C₆×C₃⋊D₄, C₃×C₂³⋊₂D₆

Smallest permutation representation of C₃×C₂³⋊₂D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	3A	3B	3C	3D	3E	4A	4B	4C	6A	···	6F	6G	···	6S	6T	···	6AG	6AH	···	6AO	12A	···	12H	12I	12J	12K	12L
order	1	2	2	2	2	2	2	2	2	2	2	3	3	3	3	3	4	4	4	6	···	6	6	···	6	6	···	6	6	···	6	12	···	12	12	12	12	12
size	1	1	1	1	2	2	4	6	6	6	6	1	1	2	2	2	4	12	12	1	···	1	2	···	2	4	···	4	6	···	6	4	···	4	12	12	12	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+	+	+								+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₄	D₆	D₆	C₃×S₃	C₃×D₄	C₃⋊D₄	C₃×D₄	S₃×C₆	S₃×C₆	C₃×C₃⋊D₄	S₃×D₄	C₃×S₃×D₄
kernel	C₃×C₂³⋊₂D₆	C₃×D₆⋊C₄	C₃×C₆.D₄	C₆×C₃⋊D₄	D₄×C₃×C₆	S₃×C₂²×C₆	C₂³⋊₂D₆	D₆⋊C₄	C₆.D₄	C₂×C₃⋊D₄	C₆×D₄	S₃×C₂³	C₆×D₄	S₃×C₆	C₆²	C₂×C₁₂	C₂²×C₆	C₂×D₄	D₆	C₂×C₆	C₂×C₆	C₂×C₄	C₂³	C₂²	C₆	C₂
# reps	1	2	1	2	1	1	2	4	2	4	2	2	1	4	2	1	2	2	8	4	4	2	4	8	2	4

Matrix representation of C₃×C₂³⋊₂D₆ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	5
0	0	0	0	0	1

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	0
0	0	0	0	0	12

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	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	9	0	0	0
0	0	0	3	0	0
0	0	0	0	12	0
0	0	0	0	10	1

1	0	0	0	0	0
0	12	0	0	0	0
0	0	0	3	0	0
0	0	9	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,12,10,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃×C₂³⋊₂D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes_2D_6

% in TeX

G:=Group("C3xC2^3:2D6");

// GroupNames label

G:=SmallGroup(288,708);

// by ID

G=gap.SmallGroup(288,708);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,9414]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^6=f^2=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,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C3×C23⋊2D6 order 288 = 25·32

Direct product of C3 and C23⋊2D6

G = C₃×C₂³⋊₂D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³⋊₂D₆