C3xC3:S3.Q8 - GroupNames

G = C₃×C₃⋊S₃.Q₈ order 432 = 2⁴·3³

Direct product of C₃ and C₃⋊S₃.Q₈

direct product, non-abelian, soluble, monomial

Aliases: C₃×C₃⋊S₃.Q₈, C₆.₂₀S₃≀C₂, C₃²⋊C₄⋊₁C₁₂, C₃³⋊₂(C₄⋊C₄), (C₃²×C₆).₂D₄, C₆.D₆.₂C₆, C₃⋊S₃.(C₃×Q₈), C₂.₂(C₃×S₃≀C₂), (C₃×C₃²⋊C₄)⋊₄C₄, C₃²⋊₁(C₃×C₄⋊C₄), (C₃×C₆).₂(C₃×D₄), (C₃×C₃⋊S₃).₁Q₈, C₃⋊S₃.₃(C₂×C₁₂), (C₂×C₃²⋊C₄).₁C₆, (C₆×C₃⋊S₃).₅C₂², (C₆×C₃²⋊C₄).₁₀C₂, (C₃×C₆.D₆).₃C₂, (C₂×C₃⋊S₃).₅(C₂×C₆), (C₃×C₃⋊S₃).₅(C₂×C₄), SmallGroup(432,575)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊S₃ — C₃×C₃⋊S₃.Q₈

Chief series

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

Lower central

C₃² — C₃⋊S₃ — C₃×C₃⋊S₃.Q₈

Upper central

C₁ — C₆

Generators and relations for C₃×C₃⋊S₃.Q₈
G = < a,b,c,d,e,f | a³=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: 428 in 96 conjugacy classes, 26 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₂×C₄, C₃², C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₄⋊C₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃×C₁₂, C₃²⋊C₄, S₃×C₆, C₂×C₃⋊S₃, C₃×C₄⋊C₄, C₃×C₃⋊S₃, C₃²×C₆, C₆.D₆, S₃×C₁₂, C₂×C₃²⋊C₄, C₃²×Dic₃, C₃×C₃²⋊C₄, C₆×C₃⋊S₃, C₃⋊S₃.Q₈, C₃×C₆.D₆, C₆×C₃²⋊C₄, C₃×C₃⋊S₃.Q₈
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, Q₈, C₁₂, C₂×C₆, C₄⋊C₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₃×C₄⋊C₄, S₃≀C₂, C₃⋊S₃.Q₈, C₃×S₃≀C₂, C₃×C₃⋊S₃.Q₈

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

Generators in S₄₈

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

(1 5 17)(3 7 19)(9 23 35)(11 21 33)(13 47 40)(15 45 38)(25 29 44)(27 31 42)

(2 18 6)(4 20 8)(10 36 24)(12 34 22)(14 37 48)(16 39 46)(26 41 30)(28 43 32)

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	4A	4B	4C	4D	4E	4F	6A	6B	6C	···	6H	6I	6J	6K	6L	12A	···	12H	12I	···	12T	12U	12V	12W	12X
order	1	2	2	2	3	3	3	···	3	4	4	4	4	4	4	6	6	6	···	6	6	6	6	6	12	···	12	12	···	12	12	12	12	12
size	1	1	9	9	1	1	4	···	4	6	6	6	6	18	18	1	1	4	···	4	9	9	9	9	6	···	6	12	···	12	18	18	18	18

54 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+						-	+			+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	Q₈	D₄	C₃×Q₈	C₃×D₄	S₃≀C₂	C₃⋊S₃.Q₈	C₃×S₃≀C₂	C₃×C₃⋊S₃.Q₈
kernel	C₃×C₃⋊S₃.Q₈	C₃×C₆.D₆	C₆×C₃²⋊C₄	C₃⋊S₃.Q₈	C₃×C₃²⋊C₄	C₆.D₆	C₂×C₃²⋊C₄	C₃²⋊C₄	C₃×C₃⋊S₃	C₃²×C₆	C₃⋊S₃	C₃×C₆	C₆	C₃	C₂	C₁
# reps	1	2	1	2	4	4	2	8	1	1	2	2	4	4	8	8

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

3	0	0	0	0	0
0	3	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	3	9	0	0
0	0	0	0	1	0
0	0	2	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	9	0
0	0	11	0	3	3

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	2	0	0
0	0	0	12	0	0
0	0	11	0	12	2
0	0	0	2	0	1

5	0	0	0	0	0
1	8	0	0	0	0
0	0	0	0	1	0
0	0	12	0	12	1
0	0	1	0	0	0
0	0	1	1	1	0

12	10	0	0	0	0
5	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	11	0	12	2
0	0	0	0	0	1

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,2,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,11,0,0,0,2,12,0,2,0,0,0,0,12,0,0,0,0,0,2,1],[5,1,0,0,0,0,0,8,0,0,0,0,0,0,0,12,1,1,0,0,0,0,0,1,0,0,1,12,0,1,0,0,0,1,0,0],[12,5,0,0,0,0,10,1,0,0,0,0,0,0,1,0,11,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,2,1] >;

C₃×C₃⋊S₃.Q₈ in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes S_3.Q_8

% in TeX

G:=Group("C3xC3:S3.Q8");

// GroupNames label

G:=SmallGroup(432,575);

// by ID

G=gap.SmallGroup(432,575);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,168,197,176,4037,3036,362,1189,1203]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=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 = C3×C3⋊S3.Q8 order 432 = 24·33

Direct product of C3 and C3⋊S3.Q8

G = C₃×C₃⋊S₃.Q₈ order 432 = 2⁴·3³

Direct product of C₃ and C₃⋊S₃.Q₈