SD16xES-(3,1) - GroupNames

G = SD₁₆×3_-¹⁺² order 432 = 2⁴·3³

Direct product of SD₁₆ and 3_-¹⁺²

direct product, metacyclic, nilpotent (class 3), monomial

Aliases: SD₁₆×3_-¹⁺², C₇₂⋊₆C₆, C₁₂.₂₂C₆², (C₉×SD₁₆)⋊C₃, (Q₈×C₉)⋊₆C₆, (C₃×C₂₄).₆C₆, (D₄×C₉).₂C₆, C₉⋊₄(C₃×SD₁₆), C₃₆.₁₈(C₂×C₆), C₂₄.₁₂(C₃×C₆), C₁₈.₁₅(C₃×D₄), C₃².(C₃×SD₁₆), (D₄×C₃²).₄C₆, C₆.₃₂(D₄×C₃²), (C₃²×SD₁₆).C₃, (Q₈×C₃²).₇C₆, C₈⋊₂(C₂×3_-¹⁺²), C₃.₃(C₃²×SD₁₆), D₄.(C₂×3_-¹⁺²), Q₈⋊₃(C₂×3_-¹⁺²), (C₈×3_-¹⁺²)⋊₆C₂, (Q₈×3_-¹⁺²)⋊₄C₂, (C₃×SD₁₆).₃C₃², C₂.₄(D₄×3_-¹⁺²), (D₄×3_-¹⁺²).₂C₂, (C₂×3_-¹⁺²).₁₅D₄, C₄.₂(C₂²×3_-¹⁺²), (C₄×3_-¹⁺²).₁₈C₂², (C₃×D₄).₇(C₃×C₆), (C₃×C₆).₃₅(C₃×D₄), (C₃×C₁₂).₂₀(C₂×C₆), (C₃×Q₈).₁₆(C₃×C₆), SmallGroup(432,220)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — SD₁₆×3_-¹⁺²

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₃×C₁₂ — C₄×3_-¹⁺² — Q₈×3_-¹⁺² — SD₁₆×3_-¹⁺²

Lower central

C₁ — C₂ — C₁₂ — SD₁₆×3_-¹⁺²

Upper central

C₁ — C₆ — C₄×3_-¹⁺² — SD₁₆×3_-¹⁺²

Generators and relations for SD₁₆×3_-¹⁺²
G = < a,b,c,d | a⁸=b²=c⁹=d³=1, bab=a³, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁴ >

Subgroups: 150 in 80 conjugacy classes, 49 normal (28 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₈, D₄, Q₈, C₉, C₃², C₁₂, C₁₂, C₂×C₆, SD₁₆, C₁₈, C₁₈, C₃×C₆, C₃×C₆, C₂₄, C₂₄, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, 3_-¹⁺², C₃₆, C₃₆, C₂×C₁₈, C₃×C₁₂, C₃×C₁₂, C₆², C₃×SD₁₆, C₃×SD₁₆, C₂×3_-¹⁺², C₂×3_-¹⁺², C₇₂, D₄×C₉, Q₈×C₉, C₃×C₂₄, D₄×C₃², Q₈×C₃², C₄×3_-¹⁺², C₄×3_-¹⁺², C₂²×3_-¹⁺², C₉×SD₁₆, C₃²×SD₁₆, C₈×3_-¹⁺², D₄×3_-¹⁺², Q₈×3_-¹⁺², SD₁₆×3_-¹⁺²
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₃², C₂×C₆, SD₁₆, C₃×C₆, C₃×D₄, 3_-¹⁺², C₆², C₃×SD₁₆, C₂×3_-¹⁺², D₄×C₃², C₂²×3_-¹⁺², C₃²×SD₁₆, D₄×3_-¹⁺², SD₁₆×3_-¹⁺²

Smallest permutation representation of SD₁₆×3_-¹⁺²
►On 72 points

Generators in S₇₂

(1 66 63 15 48 21 43 31)(2 67 55 16 49 22 44 32)(3 68 56 17 50 23 45 33)(4 69 57 18 51 24 37 34)(5 70 58 10 52 25 38 35)(6 71 59 11 53 26 39 36)(7 72 60 12 54 27 40 28)(8 64 61 13 46 19 41 29)(9 65 62 14 47 20 42 30)

(10 70)(11 71)(12 72)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 28)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 55)(45 56)

(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)

(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(46 52 49)(47 50 53)(55 61 58)(56 59 62)(64 70 67)(65 68 71)

G:=sub<Sym(72)| (1,66,63,15,48,21,43,31)(2,67,55,16,49,22,44,32)(3,68,56,17,50,23,45,33)(4,69,57,18,51,24,37,34)(5,70,58,10,52,25,38,35)(6,71,59,11,53,26,39,36)(7,72,60,12,54,27,40,28)(8,64,61,13,46,19,41,29)(9,65,62,14,47,20,42,30), (10,70)(11,71)(12,72)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,28)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,55)(45,56), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(46,52,49)(47,50,53)(55,61,58)(56,59,62)(64,70,67)(65,68,71)>;

G:=Group( (1,66,63,15,48,21,43,31)(2,67,55,16,49,22,44,32)(3,68,56,17,50,23,45,33)(4,69,57,18,51,24,37,34)(5,70,58,10,52,25,38,35)(6,71,59,11,53,26,39,36)(7,72,60,12,54,27,40,28)(8,64,61,13,46,19,41,29)(9,65,62,14,47,20,42,30), (10,70)(11,71)(12,72)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,28)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,55)(45,56), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(46,52,49)(47,50,53)(55,61,58)(56,59,62)(64,70,67)(65,68,71) );

G=PermutationGroup([[(1,66,63,15,48,21,43,31),(2,67,55,16,49,22,44,32),(3,68,56,17,50,23,45,33),(4,69,57,18,51,24,37,34),(5,70,58,10,52,25,38,35),(6,71,59,11,53,26,39,36),(7,72,60,12,54,27,40,28),(8,64,61,13,46,19,41,29),(9,65,62,14,47,20,42,30)], [(10,70),(11,71),(12,72),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,28),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,55),(45,56)], [(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),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(46,52,49),(47,50,53),(55,61,58),(56,59,62),(64,70,67),(65,68,71)]])

77 conjugacy classes

class	1	2A	2B	3A	3B	3C	3D	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	9A	···	9F	12A	12B	12C	12D	12E	12F	12G	12H	18A	···	18F	18G	···	18L	24A	24B	24C	24D	24E	24F	24G	24H	36A	···	36F	36G	···	36L	72A	···	72L
order	1	2	2	3	3	3	3	4	4	6	6	6	6	6	6	6	6	8	8	9	···	9	12	12	12	12	12	12	12	12	18	···	18	18	···	18	24	24	24	24	24	24	24	24	36	···	36	36	···	36	72	···	72
size	1	1	4	1	1	3	3	2	4	1	1	3	3	4	4	12	12	2	2	3	···	3	2	2	4	4	6	6	12	12	3	···	3	12	···	12	2	2	2	2	6	6	6	6	6	···	6	12	···	12	6	···	6

77 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	6	6
type	+	+	+	+									+
image	C₁	C₂	C₂	C₂	C₃	C₃	C₆	C₆	C₆	C₆	C₆	C₆	D₄	SD₁₆	C₃×D₄	C₃×D₄	C₃×SD₁₆	C₃×SD₁₆	3_-¹⁺²	C₂×3_-¹⁺²	C₂×3_-¹⁺²	C₂×3_-¹⁺²	D₄×3_-¹⁺²	SD₁₆×3_-¹⁺²
kernel	SD₁₆×3_-¹⁺²	C₈×3_-¹⁺²	D₄×3_-¹⁺²	Q₈×3_-¹⁺²	C₉×SD₁₆	C₃²×SD₁₆	C₇₂	D₄×C₉	Q₈×C₉	C₃×C₂₄	D₄×C₃²	Q₈×C₃²	C₂×3_-¹⁺²	3_-¹⁺²	C₁₈	C₃×C₆	C₉	C₃²	SD₁₆	C₈	D₄	Q₈	C₂	C₁
# reps	1	1	1	1	6	2	6	6	6	2	2	2	1	2	6	2	12	4	2	2	2	2	2	4

Matrix representation of SD₁₆×3_-¹⁺² ►in GL₅(𝔽₇₃)

67	6	0	0	0
67	67	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

8	0	0	0	0
0	8	0	0	0
0	0	64	16	0
0	0	68	9	1
0	0	0	72	0

8	0	0	0	0
0	8	0	0	0
0	0	1	0	0
0	0	33	64	0
0	0	5	0	8

G:=sub<GL(5,GF(73))| [67,67,0,0,0,6,67,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[8,0,0,0,0,0,8,0,0,0,0,0,64,68,0,0,0,16,9,72,0,0,0,1,0],[8,0,0,0,0,0,8,0,0,0,0,0,1,33,5,0,0,0,64,0,0,0,0,0,8] >;

SD₁₆×3_-¹⁺² in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times 3_-^{1+2}

% in TeX

G:=Group("SD16xES-(3,1)");

// GroupNames label

G:=SmallGroup(432,220);

// by ID

G=gap.SmallGroup(432,220);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,394,605,8824,4421,242]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^3=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;

// generators/relations

G = SD16×3- 1+2 order 432 = 24·33

Direct product of SD16 and 3- 1+2

G = SD₁₆×3_-¹⁺² order 432 = 2⁴·3³

Direct product of SD₁₆ and 3_-¹⁺²