C3xC3^2:5SD16 - GroupNames

G = C₃xC₃²:₅SD₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²:₅SD₁₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₃²:₅SD₁₆, C₃³:₁₁SD₁₆, C₁₂.₇₆S₃², C₁₂.₂₉(S₃xC₆), Dic₆:₁(C₃xS₃), (C₃xDic₆):₂C₆, C₆.₂₆(C₃xD₁₂), (C₃xC₆).₇₃D₁₂, C₁₂:S₃.₄C₆, (C₃xDic₆):₁₀S₃, (C₃xC₁₂).₁₇₄D₆, C₃²:₈(C₃xSD₁₆), (C₃²xC₆).₂₂D₄, (C₃²xDic₆):₂C₂, C₃²:₁₂(C₂₄:C₂), C₆.₄₄(C₃:D₁₂), (C₃²xC₁₂).₅C₂², C₃²:₁₀(Q₈:₂S₃), (C₃xC₃:C₈):₃C₆, C₃:C₈:₃(C₃xS₃), C₄.₃(C₃xS₃²), (C₃xC₃:C₈):₁₀S₃, C₃:₂(C₃xC₂₄:C₂), (C₃²xC₃:C₈):₆C₂, C₆.₃(C₃xC₃:D₄), C₃:₁(C₃xQ₈:₂S₃), (C₃xC₆).₂₁(C₃xD₄), (C₃xC₁₂).₃₉(C₂xC₆), C₂.₆(C₃xC₃:D₁₂), (C₃xC₁₂:S₃).₁C₂, (C₃xC₆).₇₂(C₃:D₄), SmallGroup(432,422)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₂ — C₃xC₃²:₅SD₁₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₃²xC₁₂ — C₃²xDic₆ — C₃xC₃²:₅SD₁₆

Lower central

C₃² — C₃xC₆ — C₃xC₁₂ — C₃xC₃²:₅SD₁₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xC₃²:₅SD₁₆
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=ebe=b^-1, cd=dc, ece=c^-1, ede=d³ >

Subgroups: 520 in 122 conjugacy classes, 36 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, D₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, SD₁₆, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, Dic₆, D₁₂, C₃xD₄, C₃xQ₈, C₃³, C₃xDic₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₂₄:C₂, Q₈:₂S₃, C₃xSD₁₆, C₃xC₃:S₃, C₃²xC₆, C₃xC₃:C₈, C₃xC₃:C₈, C₃xC₂₄, C₃xDic₆, C₃xDic₆, C₃xD₁₂, C₁₂:S₃, Q₈xC₃², C₃²xDic₃, C₃²xC₁₂, C₆xC₃:S₃, C₃²:₅SD₁₆, C₃xC₂₄:C₂, C₃xQ₈:₂S₃, C₃²xC₃:C₈, C₃²xDic₆, C₃xC₁₂:S₃, C₃xC₃²:₅SD₁₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂xC₆, SD₁₆, C₃xS₃, D₁₂, C₃:D₄, C₃xD₄, S₃², S₃xC₆, C₂₄:C₂, Q₈:₂S₃, C₃xSD₁₆, C₃:D₁₂, C₃xD₁₂, C₃xC₃:D₄, C₃xS₃², C₃²:₅SD₁₆, C₃xC₂₄:C₂, C₃xQ₈:₂S₃, C₃xC₃:D₁₂, C₃xC₃²:₅SD₁₆

Smallest permutation representation of C₃xC₃²:₅SD₁₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	6B	6C	···	6H	6I	6J	6K	6L	6M	8A	8B	12A	···	12H	12I	···	12Q	12R	···	12Y	24A	···	24P
order	1	2	2	3	3	3	···	3	3	3	3	4	4	6	6	6	···	6	6	6	6	6	6	8	8	12	···	12	12	···	12	12	···	12	24	···	24
size	1	1	36	1	1	2	···	2	4	4	4	2	12	1	1	2	···	2	4	4	4	36	36	6	6	2	···	2	4	···	4	12	···	12	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	SD₁₆	C₃xS₃	C₃xS₃	D₁₂	C₃:D₄	C₃xD₄	S₃xC₆	C₂₄:C₂	C₃xSD₁₆	C₃xD₁₂	C₃xC₃:D₄	C₃xC₂₄:C₂	S₃²	Q₈:₂S₃	C₃:D₁₂	C₃xS₃²	C₃²:₅SD₁₆	C₃xQ₈:₂S₃	C₃xC₃:D₁₂	C₃xC₃²:₅SD₁₆
kernel	C₃xC₃²:₅SD₁₆	C₃²xC₃:C₈	C₃²xDic₆	C₃xC₁₂:S₃	C₃²:₅SD₁₆	C₃xC₃:C₈	C₃xDic₆	C₁₂:S₃	C₃xC₃:C₈	C₃xDic₆	C₃²xC₆	C₃xC₁₂	C₃³	C₃:C₈	Dic₆	C₃xC₆	C₃xC₆	C₃xC₆	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 C₃xC₃²:₅SD₁₆ ►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	1
0	0	0	0	72	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

51	47	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	1	0
0	0	0	0	72	72

21	57	0	0	0	0
64	52	0	0	0	0
0	0	0	72	0	0
0	0	72	0	0	0
0	0	0	0	1	0
0	0	0	0	72	72

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[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],[51,0,0,0,0,0,47,10,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[21,64,0,0,0,0,57,52,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C₃xC₃²:₅SD₁₆ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_5{\rm SD}_{16}

% in TeX

G:=Group("C3xC3^2:5SD16");

// GroupNames label

G:=SmallGroup(432,422);

// by ID

G=gap.SmallGroup(432,422);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,92,1011,80,2028,14118]);

// Polycyclic

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

// generators/relations

G = C3xC32:5SD16 order 432 = 24·33

Direct product of C3 and C32:5SD16

G = C₃xC₃²:₅SD₁₆ order 432 = 2⁴·3³

Direct product of C₃ and C₃²:₅SD₁₆