C3xC3^2:2Q16 - GroupNames

G = C₃×C₃²⋊₂Q₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₂Q₁₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₃²⋊₂Q₁₆, C₃³⋊₄Q₁₆, C₁₂.₉₈S₃², C₁₂.₃₀(S₃×C₆), C₃²⋊₄(C₃×Q₁₆), (C₃×C₁₂).₁₁₁D₆, C₃²⋊₄C₈.₂C₆, (C₃×Dic₆).₉S₃, Dic₆.₂(C₃×S₃), (C₃×Dic₆).₅C₆, (C₃²×C₆).₂₃D₄, C₆.₂₉(D₆⋊S₃), C₃²⋊₁₀(C₃⋊Q₁₆), (C₃²×C₁₂).₆C₂², (C₃²×Dic₆).₁C₂, C₄.₁₀(C₃×S₃²), C₃⋊₂(C₃×C₃⋊Q₁₆), C₆.₉(C₃×C₃⋊D₄), (C₃×C₆).₂₂(C₃×D₄), (C₃×C₁₂).₄₀(C₂×C₆), C₂.₅(C₃×D₆⋊S₃), (C₃×C₆).₈₄(C₃⋊D₄), (C₃×C₃²⋊₄C₈).₄C₂, SmallGroup(432,423)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃×C₃²⋊₂Q₁₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — C₃²×Dic₆ — C₃×C₃²⋊₂Q₁₆

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×C₃²⋊₂Q₁₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×C₃²⋊₂Q₁₆
G = < a,b,c,d,e | a³=b³=c³=d⁸=1, e²=d⁴, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=ebe^-1=b^-1, dcd^-1=c^-1, ce=ec, ede^-1=d^-1 >

Subgroups: 304 in 110 conjugacy classes, 36 normal (16 characteristic)
C₁, C₂, C₃, C₃, C₃, C₄, C₄, C₆, C₆, C₆, C₈, Q₈, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, Q₁₆, C₃×C₆, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, C₃×Q₈, C₃³, C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, C₃×C₁₂, C₃⋊Q₁₆, C₃×Q₁₆, C₃²×C₆, C₃×C₃⋊C₈, C₃²⋊₄C₈, C₃×Dic₆, C₃×Dic₆, Q₈×C₃², C₃²×Dic₃, C₃²×C₁₂, C₃²⋊₂Q₁₆, C₃×C₃⋊Q₁₆, C₃×C₃²⋊₄C₈, C₃²×Dic₆, C₃×C₃²⋊₂Q₁₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, Q₁₆, C₃×S₃, C₃⋊D₄, C₃×D₄, S₃², S₃×C₆, C₃⋊Q₁₆, C₃×Q₁₆, D₆⋊S₃, C₃×C₃⋊D₄, C₃×S₃², C₃²⋊₂Q₁₆, C₃×C₃⋊Q₁₆, C₃×D₆⋊S₃, C₃×C₃²⋊₂Q₁₆

Smallest permutation representation of C₃×C₃²⋊₂Q₁₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

63 conjugacy classes

class	1	2	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	6A	6B	6C	···	6H	6I	6J	6K	8A	8B	12A	12B	12C	···	12N	12O	···	12AD	24A	24B	24C	24D
order	1	2	3	3	3	···	3	3	3	3	4	4	4	6	6	6	···	6	6	6	6	8	8	12	12	12	···	12	12	···	12	24	24	24	24
size	1	1	1	1	2	···	2	4	4	4	2	12	12	1	1	2	···	2	4	4	4	18	18	2	2	4	···	4	12	···	12	18	18	18	18

63 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+				+	+	+	-							+	-	-
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₆	Q₁₆	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	C₃×Q₁₆	C₃×C₃⋊D₄	S₃²	C₃⋊Q₁₆	D₆⋊S₃	C₃×S₃²	C₃²⋊₂Q₁₆	C₃×C₃⋊Q₁₆	C₃×D₆⋊S₃	C₃×C₃²⋊₂Q₁₆
kernel	C₃×C₃²⋊₂Q₁₆	C₃×C₃²⋊₄C₈	C₃²×Dic₆	C₃²⋊₂Q₁₆	C₃²⋊₄C₈	C₃×Dic₆	C₃×Dic₆	C₃²×C₆	C₃×C₁₂	C₃³	Dic₆	C₃×C₆	C₃×C₆	C₁₂	C₃²	C₆	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	2	2	2	4	2	1	2	2	4	4	2	4	4	8	1	2	1	2	2	4	2	4

Matrix representation of C₃×C₃²⋊₂Q₁₆ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	64	0	0	0
0	0	0	64	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	72
0	0	0	0	1	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

22	0	0	0	0	0
0	10	0	0	0	0
0	0	0	72	0	0
0	0	72	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

0	1	0	0	0	0
72	0	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₃×C₃²⋊₂Q₁₆ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2Q_{16}

% in TeX

G:=Group("C3xC3^2:2Q16");

// GroupNames label

G:=SmallGroup(432,423);

// by ID

G=gap.SmallGroup(432,423);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C3×C32⋊2Q16 order 432 = 24·33

Direct product of C3 and C32⋊2Q16

G = C₃×C₃²⋊₂Q₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₂Q₁₆