C6xC3^2:2Q8 - GroupNames

G = C₆×C₃²⋊₂Q₈ order 432 = 2⁴·3³

Direct product of C₆ and C₃²⋊₂Q₈

direct product, metabelian, supersoluble, monomial

Aliases: C₆×C₃²⋊₂Q₈, C₆².₁₁₂D₆, C₆⋊₁(C₃×Dic₆), (C₃×C₆)⋊₇Dic₆, C₃⋊₂(C₆×Dic₆), (C₃²×C₆)⋊₃Q₈, C₃²⋊₆(C₆×Q₈), C₃³⋊₁₃(C₂×Q₈), C₆².₂₈(C₂×C₆), Dic₃.₇(S₃×C₆), (C₆×Dic₃).₈C₆, (C₃×Dic₃).₄₈D₆, (C₆×Dic₃).₁₉S₃, C₃²⋊₁₄(C₂×Dic₆), (C₃²×C₆).₃₅C₂³, (C₃×C₆²).₂₂C₂², (C₃²×Dic₃).₂₅C₂², C₂.₁₆(S₃²×C₆), (C₂×C₆).₇₅S₃², C₆.₁₆(S₃×C₂×C₆), (C₃×C₆)⋊₃(C₃×Q₈), C₆.₁₁₉(C₂×S₃²), (C₂×C₆).₃₀(S₃×C₆), C₂².₁₂(C₃×S₃²), (Dic₃×C₃×C₆).₉C₂, (C₂×C₃⋊Dic₃).₁₂C₆, C₃⋊Dic₃.₂₀(C₂×C₆), (C₆×C₃⋊Dic₃).₁₈C₂, (C₃×C₆).₂₆(C₂²×C₆), (C₃×Dic₃).₇(C₂×C₆), (C₂×Dic₃).₃(C₃×S₃), (C₃×C₆).₁₄₀(C₂²×S₃), (C₃×C₃⋊Dic₃).₅₇C₂², SmallGroup(432,657)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₆×C₃²⋊₂Q₈

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃²×C₆ — C₃²×Dic₃ — C₃×C₃²⋊₂Q₈ — C₆×C₃²⋊₂Q₈

Lower central

C₃² — C₃×C₆ — C₆×C₃²⋊₂Q₈

Upper central

C₁ — C₂×C₆

Generators and relations for C₆×C₃²⋊₂Q₈
G = < a,b,c,d,e | a⁶=b³=c³=d⁴=1, e²=d², ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=b^-1, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d^-1 >

Subgroups: 576 in 210 conjugacy classes, 80 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₂×C₄, Q₈, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂×Q₈, C₃×C₆, C₃×C₆, C₃×C₆, Dic₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₃×Q₈, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₆², C₆², C₆², C₂×Dic₆, C₆×Q₈, C₃²×C₆, C₃²×C₆, C₃²⋊₂Q₈, C₃×Dic₆, C₆×Dic₃, C₆×Dic₃, C₂×C₃⋊Dic₃, C₆×C₁₂, C₃²×Dic₃, C₃×C₃⋊Dic₃, C₃×C₆², C₂×C₃²⋊₂Q₈, C₆×Dic₆, C₃×C₃²⋊₂Q₈, Dic₃×C₃×C₆, C₆×C₃⋊Dic₃, C₆×C₃²⋊₂Q₈
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, C₂³, D₆, C₂×C₆, C₂×Q₈, C₃×S₃, Dic₆, C₃×Q₈, C₂²×S₃, C₂²×C₆, S₃², S₃×C₆, C₂×Dic₆, C₆×Q₈, C₃²⋊₂Q₈, C₃×Dic₆, C₂×S₃², S₃×C₂×C₆, C₃×S₃², C₂×C₃²⋊₂Q₈, C₆×Dic₆, C₃×C₃²⋊₂Q₈, S₃²×C₆, C₆×C₃²⋊₂Q₈

Smallest permutation representation of C₆×C₃²⋊₂Q₈
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

90 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	4D	4E	4F	6A	···	6F	6G	···	6X	6Y	···	6AG	12A	···	12AF	12AG	12AH	12AI	12AJ
order	1	2	2	2	3	3	3	···	3	3	3	3	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	12	···	12	12	12	12	12
size	1	1	1	1	1	1	2	···	2	4	4	4	6	6	6	6	18	18	1	···	1	2	···	2	4	···	4	6	···	6	18	18	18	18

90 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+					+	-	+	+		-					+	-	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	Q₈	D₆	D₆	C₃×S₃	Dic₆	C₃×Q₈	S₃×C₆	S₃×C₆	C₃×Dic₆	S₃²	C₃²⋊₂Q₈	C₂×S₃²	C₃×S₃²	C₃×C₃²⋊₂Q₈	S₃²×C₆
kernel	C₆×C₃²⋊₂Q₈	C₃×C₃²⋊₂Q₈	Dic₃×C₃×C₆	C₆×C₃⋊Dic₃	C₂×C₃²⋊₂Q₈	C₃²⋊₂Q₈	C₆×Dic₃	C₂×C₃⋊Dic₃	C₆×Dic₃	C₃²×C₆	C₃×Dic₃	C₆²	C₂×Dic₃	C₃×C₆	C₃×C₆	Dic₃	C₂×C₆	C₆	C₂×C₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	4	2	1	2	8	4	2	2	2	4	2	4	8	4	8	4	16	1	2	1	2	4	2

Matrix representation of C₆×C₃²⋊₂Q₈ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	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

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	0	12
0	0	0	0	1	12

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

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

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

C₆×C₃²⋊₂Q₈ in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_2Q_8

% in TeX

G:=Group("C6xC3^2:2Q8");

// GroupNames label

G:=SmallGroup(432,657);

// by ID

G=gap.SmallGroup(432,657);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,365,176,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;

// generators/relations

G = C6×C32⋊2Q8 order 432 = 24·33

Direct product of C6 and C32⋊2Q8

G = C₆×C₃²⋊₂Q₈ order 432 = 2⁴·3³

Direct product of C₆ and C₃²⋊₂Q₈