Dic3:C8 - GroupNames

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

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

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

Dic₃⋊C₈ is a maximal subgroup of
C₈×Dic₆ C₂₄⋊₁₂Q₈ C₄².₂₈₂D₆ C₄².₂₄₃D₆ C₂₄⋊Q₈ C₄².₁₈₂D₆ C₄².₁₈₅D₆ Dic₃.₅M₄(2) Dic₃.M₄(2) C₂₄⋊C₄⋊C₂ C₃⋊D₄⋊C₈ D₆⋊₂M₄(2) Dic₃⋊M₄(2) C₃⋊C₈⋊₂₆D₄ Dic₃.D₈ D₄⋊Dic₆ Dic₆⋊₂D₄ D₄.Dic₆ D₄.₂Dic₆ Dic₆.D₄ D₁₂⋊₃D₄ D₁₂.D₄ Q₈⋊₂Dic₆ Q₈⋊₃Dic₆ Dic₃⋊Q₁₆ Q₈.₃Dic₆ Dic₆.₁₁D₄ Q₈.₄Dic₆ Dic₃⋊SD₁₆ D₁₂.₁₂D₄ C₄².₂₇D₆ Dic₆⋊C₈ C₄².₁₉₈D₆ S₃×C₄⋊C₈ C₁₂⋊M₄(2) C₄².₃₀D₆ Dic₆⋊Q₈ Dic₆.Q₈ D₁₂⋊Q₈ D₁₂.Q₈ Dic₃.Q₁₆ Dic₆.₂Q₈ D₁₂⋊₂Q₈ D₁₂.₂Q₈ Dic₃⋊C₈⋊C₂ C₈×C₃⋊D₄ C₂₄⋊₃₃D₄ Dic₃⋊₄M₄(2) C₁₂.₈₈(C₂×Q₈) C₂₄⋊D₄ C₂₄⋊₂₁D₄ Dic₃⋊D₈ (C₆×D₈).C₂ Dic₃⋊₃SD₁₆ Dic₃⋊₅SD₁₆ (C₃×D₄).D₄ (C₃×Q₈).D₄ Dic₃⋊₃Q₁₆ (C₂×Q₁₆)⋊S₃ Dic₉⋊C₈ C₁₂.₈₁D₁₂ C₁₂.₁₅Dic₆ C₁₂.₃₀Dic₆ C₆₀.₁₄Q₈ C₆₀.₁₅Q₈ C₆₀.₂₆Q₈ C₃₀.₄M₄(2) Dic₁₅⋊C₈
Dic₃⋊C₈ is a maximal quotient of
C₁₂.₅₃D₈ C₁₂.₃₉SD₁₆ Dic₃⋊C₁₆ C₂₄.₉₇D₄ (C₂×C₂₄)⋊₅C₄ Dic₉⋊C₈ C₁₂.₈₁D₁₂ C₁₂.₁₅Dic₆ C₁₂.₃₀Dic₆ C₆₀.₁₄Q₈ C₆₀.₁₅Q₈ C₆₀.₂₆Q₈ C₃₀.₄M₄(2) Dic₁₅⋊C₈

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+			+	+	-	+		-
image	C₁	C₂	C₂	C₂	C₄	C₈	S₃	D₄	Q₈	D₆	M₄(2)	Dic₆	C₃⋊D₄	C₄×S₃	S₃×C₈	C₈⋊S₃
kernel	Dic₃⋊C₈	C₂×C₃⋊C₈	C₄×Dic₃	C₂×C₂₄	C₂×Dic₃	Dic₃	C₂×C₈	C₁₂	C₁₂	C₂×C₄	C₆	C₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	4	8	1	1	1	1	2	2	2	2	4	4

Matrix representation of Dic₃⋊C₈ ►in GL₄(𝔽₇₃) generated by

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

23	32	0	0
61	50	0	0
0	0	8	26
0	0	34	65

0	1	0	0
46	0	0	0
0	0	43	60
0	0	13	30

G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,1,0,0,72,0],[23,61,0,0,32,50,0,0,0,0,8,34,0,0,26,65],[0,46,0,0,1,0,0,0,0,0,43,13,0,0,60,30] >;

Dic₃⋊C₈ in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes C_8

% in TeX

G:=Group("Dic3:C8");

// GroupNames label

G:=SmallGroup(96,21);

// by ID

G=gap.SmallGroup(96,21);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,86,2309]);

// Polycyclic

G:=Group<a,b,c|a^6=c^8=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;

// generators/relations

Export

Subgroup lattice of Dic₃⋊C₈ in TeX

G = Dic3⋊C8 order 96 = 25·3

The semidirect product of Dic3 and C8 acting via C8/C4=C2

G = Dic₃⋊C₈ order 96 = 2⁵·3

The semidirect product of Dic₃ and C₈ acting via C₈/C₄=C₂