C3xC3^2:3Q16 - GroupNames

G = C₃×C₃²⋊₃Q₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₃Q₁₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₃²⋊₃Q₁₆, C₃³⋊₅Q₁₆, C₃²⋊₉Dic₁₂, C₁₂.₇₇S₃², C₁₂.₃₁(S₃×C₆), C₃⋊₂(C₃×Dic₁₂), (C₃×C₆).₇₄D₁₂, C₆.₂₇(C₃×D₁₂), C₃²⋊₅(C₃×Q₁₆), (C₃×C₁₂).₁₇₅D₆, Dic₆.₁(C₃×S₃), (C₃×Dic₆).₆C₆, (C₃²×C₆).₂₄D₄, C₃²⋊₄Q₈.₄C₆, (C₃×Dic₆).₁₂S₃, C₃²⋊₈(C₃⋊Q₁₆), C₆.₄₅(C₃⋊D₁₂), (C₃²×C₁₂).₇C₂², (C₃²×Dic₆).₂C₂, C₃⋊C₈.(C₃×S₃), C₄.₄(C₃×S₃²), (C₃×C₃⋊C₈).₁C₆, (C₃×C₃⋊C₈).₆S₃, C₃⋊₁(C₃×C₃⋊Q₁₆), C₆.₄(C₃×C₃⋊D₄), (C₃×C₆).₂₃(C₃×D₄), (C₃²×C₃⋊C₈).₂C₂, (C₃×C₁₂).₄₁(C₂×C₆), C₂.₇(C₃×C₃⋊D₁₂), (C₃×C₆).₇₃(C₃⋊D₄), (C₃×C₃²⋊₄Q₈).₁C₂, SmallGroup(432,424)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃×C₃²⋊₃Q₁₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — C₃²×Dic₆ — C₃×C₃²⋊₃Q₁₆

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×C₃²⋊₃Q₁₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×C₃²⋊₃Q₁₆
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=b^-1, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d^-1 >

Subgroups: 344 in 110 conjugacy classes, 36 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₄, C₆, C₆, C₈, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, Q₁₆, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₃×Q₈, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, Dic₁₂, C₃⋊Q₁₆, C₃×Q₁₆, C₃²×C₆, C₃×C₃⋊C₈, C₃×C₃⋊C₈, C₃×C₂₄, C₃×Dic₆, C₃×Dic₆, C₃²⋊₄Q₈, Q₈×C₃², C₃²×Dic₃, C₃×C₃⋊Dic₃, C₃²×C₁₂, C₃²⋊₃Q₁₆, C₃×Dic₁₂, C₃×C₃⋊Q₁₆, C₃²×C₃⋊C₈, C₃²×Dic₆, C₃×C₃²⋊₄Q₈, C₃×C₃²⋊₃Q₁₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, Q₁₆, C₃×S₃, D₁₂, C₃⋊D₄, C₃×D₄, S₃², S₃×C₆, Dic₁₂, C₃⋊Q₁₆, C₃×Q₁₆, C₃⋊D₁₂, C₃×D₁₂, C₃×C₃⋊D₄, C₃×S₃², C₃²⋊₃Q₁₆, C₃×Dic₁₂, C₃×C₃⋊Q₁₆, C₃×C₃⋊D₁₂, C₃×C₃²⋊₃Q₁₆

Smallest permutation representation of C₃×C₃²⋊₃Q₁₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 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	···	12H	12I	···	12Q	12R	···	12Y	12Z	12AA	24A	···	24P
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	12	12	24	···	24
size	1	1	1	1	2	···	2	4	4	4	2	12	36	1	1	2	···	2	4	4	4	6	6	2	···	2	4	···	4	12	···	12	36	36	6	···	6

72 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+					+	+	+	+	-			+				-					+	-	+		-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	S₃	D₄	D₆	Q₁₆	C₃×S₃	C₃×S₃	D₁₂	C₃⋊D₄	C₃×D₄	S₃×C₆	Dic₁₂	C₃×Q₁₆	C₃×D₁₂	C₃×C₃⋊D₄	C₃×Dic₁₂	S₃²	C₃⋊Q₁₆	C₃⋊D₁₂	C₃×S₃²	C₃²⋊₃Q₁₆	C₃×C₃⋊Q₁₆	C₃×C₃⋊D₁₂	C₃×C₃²⋊₃Q₁₆
kernel	C₃×C₃²⋊₃Q₁₆	C₃²×C₃⋊C₈	C₃²×Dic₆	C₃×C₃²⋊₄Q₈	C₃²⋊₃Q₁₆	C₃×C₃⋊C₈	C₃×Dic₆	C₃²⋊₄Q₈	C₃×C₃⋊C₈	C₃×Dic₆	C₃²×C₆	C₃×C₁₂	C₃³	C₃⋊C₈	Dic₆	C₃×C₆	C₃×C₆	C₃×C₆	C₁₂	C₃²	C₃²	C₆	C₆	C₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	8	1	1	1	2	2	2	2	4

Matrix representation of C₃×C₃²⋊₃Q₁₆ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	8	0	0	0
0	0	0	8	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	0	72	0	0
0	0	1	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

22	59	0	0	0	0
0	10	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

56	70	0	0	0	0
48	17	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,59,10,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[56,48,0,0,0,0,70,17,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃×C₃²⋊₃Q₁₆ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_3Q_{16}

% in TeX

G:=Group("C3xC3^2:3Q16");

// GroupNames label

G:=SmallGroup(432,424);

// by ID

G=gap.SmallGroup(432,424);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,2028,14118]);

// 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=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 = C3×C32⋊3Q16 order 432 = 24·33

Direct product of C3 and C32⋊3Q16

G = C₃×C₃²⋊₃Q₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²⋊₃Q₁₆