C3:C32 - GroupNames

(1 33 88)(2 89 34)(3 35 90)(4 91 36)(5 37 92)(6 93 38)(7 39 94)(8 95 40)(9 41 96)(10 65 42)(11 43 66)(12 67 44)(13 45 68)(14 69 46)(15 47 70)(16 71 48)(17 49 72)(18 73 50)(19 51 74)(20 75 52)(21 53 76)(22 77 54)(23 55 78)(24 79 56)(25 57 80)(26 81 58)(27 59 82)(28 83 60)(29 61 84)(30 85 62)(31 63 86)(32 87 64)

(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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,33,88)(2,89,34)(3,35,90)(4,91,36)(5,37,92)(6,93,38)(7,39,94)(8,95,40)(9,41,96)(10,65,42)(11,43,66)(12,67,44)(13,45,68)(14,69,46)(15,47,70)(16,71,48)(17,49,72)(18,73,50)(19,51,74)(20,75,52)(21,53,76)(22,77,54)(23,55,78)(24,79,56)(25,57,80)(26,81,58)(27,59,82)(28,83,60)(29,61,84)(30,85,62)(31,63,86)(32,87,64), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)>;

G:=Group( (1,33,88)(2,89,34)(3,35,90)(4,91,36)(5,37,92)(6,93,38)(7,39,94)(8,95,40)(9,41,96)(10,65,42)(11,43,66)(12,67,44)(13,45,68)(14,69,46)(15,47,70)(16,71,48)(17,49,72)(18,73,50)(19,51,74)(20,75,52)(21,53,76)(22,77,54)(23,55,78)(24,79,56)(25,57,80)(26,81,58)(27,59,82)(28,83,60)(29,61,84)(30,85,62)(31,63,86)(32,87,64), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96) );

G=PermutationGroup([[(1,33,88),(2,89,34),(3,35,90),(4,91,36),(5,37,92),(6,93,38),(7,39,94),(8,95,40),(9,41,96),(10,65,42),(11,43,66),(12,67,44),(13,45,68),(14,69,46),(15,47,70),(16,71,48),(17,49,72),(18,73,50),(19,51,74),(20,75,52),(21,53,76),(22,77,54),(23,55,78),(24,79,56),(25,57,80),(26,81,58),(27,59,82),(28,83,60),(29,61,84),(30,85,62),(31,63,86),(32,87,64)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)]])

C₃⋊C₃₂ is a maximal subgroup of
S₃×C₃₂ C₉₆⋊C₂ C₃⋊M₆(2) C₃⋊D₃₂ D₁₆.S₃ C₃⋊SD₆₄ C₃⋊Q₆₄ C₉⋊C₃₂ C₄₈.S₃ C₁₅⋊₃C₃₂ C₁₅⋊C₃₂
C₃⋊C₃₂ is a maximal quotient of
C₃⋊C₆₄ C₉⋊C₃₂ C₄₈.S₃ C₁₅⋊₃C₃₂ C₁₅⋊C₃₂

48 conjugacy classes

class	1	2	3	4A	4B	6	8A	8B	8C	8D	12A	12B	16A	···	16H	24A	24B	24C	24D	32A	···	32P	48A	···	48H
order	1	2	3	4	4	6	8	8	8	8	12	12	16	···	16	24	24	24	24	32	···	32	48	···	48
size	1	1	2	1	1	2	1	1	1	1	2	2	1	···	1	2	2	2	2	3	···	3	2	···	2

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2
type	+	+					+	-
image	C₁	C₂	C₄	C₈	C₁₆	C₃₂	S₃	Dic₃	C₃⋊C₈	C₃⋊C₁₆	C₃⋊C₃₂
kernel	C₃⋊C₃₂	C₄₈	C₂₄	C₁₂	C₆	C₃	C₁₆	C₈	C₄	C₂	C₁
# reps	1	1	2	4	8	16	1	1	2	4	8

Matrix representation of C₃⋊C₃₂ ►in GL₂(𝔽₁₇) generated by

2	14
8	14

0	11
1	0

G:=sub<GL(2,GF(17))| [2,8,14,14],[0,1,11,0] >;

C₃⋊C₃₂ in GAP, Magma, Sage, TeX

C_3\rtimes C_{32}

% in TeX

G:=Group("C3:C32");

// GroupNames label

G:=SmallGroup(96,1);

// by ID

G=gap.SmallGroup(96,1);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,12,31,50,69,2309]);

// Polycyclic

G:=Group<a,b|a^3=b^32=1,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of C₃⋊C₃₂ in TeX

G = C3⋊C32 order 96 = 25·3

The semidirect product of C3 and C32 acting via C32/C16=C2

G = C₃⋊C₃₂ order 96 = 2⁵·3

The semidirect product of C₃ and C₃₂ acting via C₃₂/C₁₆=C₂