C2xC3^3:5Q8 - GroupNames

G = C₂×C₃³⋊₅Q₈ order 432 = 2⁴·3³

Direct product of C₂ and C₃³⋊₅Q₈

direct product, metabelian, supersoluble, monomial

Aliases: C₂×C₃³⋊₅Q₈, C₆².₉₈D₆, (C₃×C₆)⋊₆Dic₆, (C₃²×C₆)⋊₅Q₈, C₃³⋊₁₅(C₂×Q₈), C₆⋊₂(C₃²⋊₂Q₈), C₃⋊Dic₃.₄₈D₆, C₃²⋊₁₁(C₂×Dic₆), (C₃²×C₆).₇₃C₂³, (C₃×C₆²).₃₆C₂², C₂².₇(C₃²⋊₄D₆), (C₂×C₆).₆₅S₃², C₆.₁₀₂(C₂×S₃²), C₃⋊₃(C₂×C₃²⋊₂Q₈), (C₆×C₃⋊Dic₃).₁₀C₂, (C₂×C₃⋊Dic₃).₁₃S₃, C₂.₉(C₂×C₃²⋊₄D₆), (C₃×C₆).₁₂₃(C₂²×S₃), (C₃×C₃⋊Dic₃).₄₇C₂², SmallGroup(432,695)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₆ — C₂×C₃³⋊₅Q₈

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃×C₃⋊Dic₃ — C₃³⋊₅Q₈ — C₂×C₃³⋊₅Q₈

Lower central

C₃³ — C₃²×C₆ — C₂×C₃³⋊₅Q₈

Upper central

C₁ — C₂²

Generators and relations for C₂×C₃³⋊₅Q₈
G = < a,b,c,d,e,f | a²=b³=c³=d³=e⁴=1, f²=e², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=fbf^-1=b^-1, cd=dc, ece^-1=c^-1, cf=fc, de=ed, fdf^-1=d^-1, fef^-1=e^-1 >

Subgroups: 856 in 210 conjugacy classes, 63 normal (7 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₂×C₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×Q₈, C₃×C₆, C₃×C₆, Dic₆, C₂×Dic₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₆², C₆², C₂×Dic₆, C₃²×C₆, C₃²×C₆, C₃²⋊₂Q₈, C₆×Dic₃, C₂×C₃⋊Dic₃, C₃×C₃⋊Dic₃, C₃×C₆², C₂×C₃²⋊₂Q₈, C₃³⋊₅Q₈, C₆×C₃⋊Dic₃, C₂×C₃³⋊₅Q₈
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂×Q₈, Dic₆, C₂²×S₃, S₃², C₂×Dic₆, C₃²⋊₂Q₈, C₂×S₃², C₃²⋊₄D₆, C₂×C₃²⋊₂Q₈, C₃³⋊₅Q₈, C₂×C₃²⋊₄D₆, C₂×C₃³⋊₅Q₈

Smallest permutation representation of C₂×C₃³⋊₅Q₈
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim	1	1	1	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	-	+	+	-	+	-	+
image	C₁	C₂	C₂	S₃	Q₈	D₆	D₆	Dic₆	S₃²	C₃²⋊₂Q₈	C₂×S₃²	C₃²⋊₄D₆	C₃³⋊₅Q₈	C₂×C₃²⋊₄D₆
kernel	C₂×C₃³⋊₅Q₈	C₃³⋊₅Q₈	C₆×C₃⋊Dic₃	C₂×C₃⋊Dic₃	C₃²×C₆	C₃⋊Dic₃	C₆²	C₃×C₆	C₂×C₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	4	3	3	2	6	3	12	3	6	3	2	4	2

Matrix representation of C₂×C₃³⋊₅Q₈ ►in GL₆(𝔽₁₃)

1	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	1	0	0	0	0
0	0	0	12	0	0
0	0	1	12	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
12	12	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	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	12
0	0	0	0	1	0

1	0	0	0	0	0
12	12	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	3	6
0	0	0	0	7	10

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	5	5
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,5,8] >;

C₂×C₃³⋊₅Q₈ in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_5Q_8

% in TeX

G:=Group("C2xC3^3:5Q8");

// GroupNames label

G:=SmallGroup(432,695);

// by ID

G=gap.SmallGroup(432,695);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,64,1124,571,2028,14118]);

// Polycyclic

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

// generators/relations

G = C2×C33⋊5Q8 order 432 = 24·33

Direct product of C2 and C33⋊5Q8

G = C₂×C₃³⋊₅Q₈ order 432 = 2⁴·3³

Direct product of C₂ and C₃³⋊₅Q₈