C9xSD16 - GroupNames

(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)

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

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

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

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

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

C₉×SD₁₆ is a maximal subgroup of D₇₂⋊C₂ SD₁₆⋊D₉ SD₁₆⋊₃D₉

63 conjugacy classes

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

63 irreducible representations

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

Matrix representation of C₉×SD₁₆ ►in GL₂(𝔽₁₉) generated by

9	0
0	9

0	5
4	6

1	11
0	18

G:=sub<GL(2,GF(19))| [9,0,0,9],[0,4,5,6],[1,0,11,18] >;

C₉×SD₁₆ in GAP, Magma, Sage, TeX

C_9\times {\rm SD}_{16}

% in TeX

G:=Group("C9xSD16");

// GroupNames label

G:=SmallGroup(144,26);

// by ID

G=gap.SmallGroup(144,26);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,432,169,122,2019,1017,165]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₉×SD₁₆ in TeX

G = C9×SD16 order 144 = 24·32

Direct product of C9 and SD16

G = C₉×SD₁₆ order 144 = 2⁴·3²

Direct product of C₉ and SD₁₆