C2xC3:S3.Q8 - GroupNames

G = C₂×C₃⋊S₃.Q₈ order 288 = 2⁵·3²

Direct product of C₂ and C₃⋊S₃.Q₈

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₃⋊S₃.Q₈, C₆².₁₀D₄, C₂².₁₃S₃≀C₂, C₆.D₆.₁₄C₂², C₃²⋊₁(C₂×C₄⋊C₄), C₃⋊S₃⋊₁(C₄⋊C₄), (C₃×C₆)⋊₁(C₄⋊C₄), C₂.₃(C₂×S₃≀C₂), (C₂×C₃²⋊C₄)⋊₄C₄, C₃²⋊C₄⋊₂(C₂×C₄), C₃⋊S₃.₄(C₂×Q₈), (C₂×C₃⋊S₃).₃Q₈, (C₂×C₃⋊S₃).₃₆D₄, (C₃×C₆).₁₅(C₂×D₄), C₃⋊S₃.₅(C₂²×C₄), (C₂×C₃⋊S₃).₉C₂³, (C₂×C₆.D₆).₈C₂, (C₂²×C₃²⋊C₄).₅C₂, (C₂×C₃²⋊C₄).₁₆C₂², (C₂²×C₃⋊S₃).₅₁C₂², (C₂×C₃⋊S₃).₁₈(C₂×C₄), SmallGroup(288,882)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊S₃ — C₂×C₃⋊S₃.Q₈

Chief series

C₁ — C₃² — C₃⋊S₃ — C₂×C₃⋊S₃ — C₆.D₆ — C₃⋊S₃.Q₈ — C₂×C₃⋊S₃.Q₈

Lower central

C₃² — C₃⋊S₃ — C₂×C₃⋊S₃.Q₈

Upper central

C₁ — C₂²

Generators and relations for C₂×C₃⋊S₃.Q₈
G = < a,b,c,d,e,f | a²=b³=c³=d²=e⁴=1, f²=e², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=ece^-1=b^-1, ebe^-1=dcd=fcf^-1=c^-1, bf=fb, de=ed, df=fd, fef^-1=de^-1 >

Subgroups: 688 in 146 conjugacy classes, 43 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₂×C₄, C₂³, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₄⋊C₄, C₂²×C₄, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂×C₄⋊C₄, C₃×Dic₃, C₃²⋊C₄, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², S₃×C₂×C₄, C₆.D₆, C₆.D₆, C₆×Dic₃, C₂×C₃²⋊C₄, C₂²×C₃⋊S₃, C₃⋊S₃.Q₈, C₂×C₆.D₆, C₂²×C₃²⋊C₄, C₂×C₃⋊S₃.Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×C₄⋊C₄, S₃≀C₂, C₃⋊S₃.Q₈, C₂×S₃≀C₂, C₂×C₃⋊S₃.Q₈

Smallest permutation representation of C₂×C₃⋊S₃.Q₈
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	···	4H	4I	4J	4K	4L	6A	···	6F	12A	···	12H
order	1	2	2	2	2	2	2	2	3	3	4	···	4	4	4	4	4	6	···	6	12	···	12
size	1	1	1	1	9	9	9	9	4	4	6	···	6	18	18	18	18	4	···	4	12	···	12

36 irreducible representations

dim	1	1	1	1	1	2	2	2	4	4	4
type	+	+	+	+		+	-	+	+		+
image	C₁	C₂	C₂	C₂	C₄	D₄	Q₈	D₄	S₃≀C₂	C₃⋊S₃.Q₈	C₂×S₃≀C₂
kernel	C₂×C₃⋊S₃.Q₈	C₃⋊S₃.Q₈	C₂×C₆.D₆	C₂²×C₃²⋊C₄	C₂×C₃²⋊C₄	C₂×C₃⋊S₃	C₂×C₃⋊S₃	C₆²	C₂²	C₂	C₂
# reps	1	4	2	1	8	1	2	1	4	8	4

Matrix representation of C₂×C₃⋊S₃.Q₈ ►in GL₆(𝔽₁₃)

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	12	1	0	0
0	0	12	0	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

12	0	0	0	0	0
0	12	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

0	8	0	0	0	0
5	0	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	0	5
0	0	0	0	5	0

6	2	0	0	0	0
2	7	0	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(13))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[6,2,0,0,0,0,2,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0] >;

C₂×C₃⋊S₃.Q₈ in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes S_3.Q_8

% in TeX

G:=Group("C2xC3:S3.Q8");

// GroupNames label

G:=SmallGroup(288,882);

// by ID

G=gap.SmallGroup(288,882);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,2693,2028,362,797,1203]);

// Polycyclic

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

// generators/relations

G = C2×C3⋊S3.Q8 order 288 = 25·32

Direct product of C2 and C3⋊S3.Q8

G = C₂×C₃⋊S₃.Q₈ order 288 = 2⁵·3²

Direct product of C₂ and C₃⋊S₃.Q₈