D6:C16 - GroupNames

G = D₆⋊C₁₆ order 192 = 2⁶·3

The semidirect product of D₆ and C₁₆ acting via C₁₆/C₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₆⋊C₁₆, C₈.₃₀D₁₂, C₂₄.₉₆D₄, C₆.₃M₅(2), C₁₂.₁₇M₄(2), (C₂×C₄₈)⋊₁C₂, (C₂×C₁₆)⋊₁S₃, C₂.₅(S₃×C₁₆), C₆.₅(C₂×C₁₆), C₃⋊₁(C₂²⋊C₁₆), C₂.₁(D₆⋊C₈), (C₂×C₈).₃₃₅D₆, C₄.₄₀(D₆⋊C₄), C₈.₄₉(C₃⋊D₄), C₆.₇(C₂²⋊C₈), (C₂²×S₃).₂C₈, C₂.₃(D₆.C₈), C₂².₁₁(S₃×C₈), (C₂×Dic₃).₄C₈, C₄.₁₅(C₈⋊S₃), C₁₂.₅₅(C₂²⋊C₄), (C₂×C₂₄).₄₂₀C₂², (C₂×C₃⋊C₁₆)⋊₉C₂, (S₃×C₂×C₈).₉C₂, (C₂×C₃⋊C₈).₁₄C₄, (S₃×C₂×C₄).₁₁C₄, (C₂×C₆).₁₂(C₂×C₈), (C₂×C₄).₁₆₉(C₄×S₃), (C₂×C₁₂).₂₄₃(C₂×C₄), SmallGroup(192,66)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — D₆⋊C₁₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂₄ — C₂×C₂₄ — S₃×C₂×C₈ — D₆⋊C₁₆

Lower central

C₃ — C₆ — D₆⋊C₁₆

Upper central

C₁ — C₂×C₈ — C₂×C₁₆

Generators and relations for D₆⋊C₁₆
G = < a,b,c | a⁶=b²=c¹⁶=1, bab=a^-1, ac=ca, cbc^-1=a³b >

Subgroups: 152 in 66 conjugacy classes, 33 normal (31 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₁₆, C₂×C₈, C₂×C₈, C₂²×C₄, C₃⋊C₈, C₂₄, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂×C₁₆, C₂×C₁₆, C₂²×C₈, C₃⋊C₁₆, C₄₈, S₃×C₈, C₂×C₃⋊C₈, C₂×C₂₄, S₃×C₂×C₄, C₂²⋊C₁₆, C₂×C₃⋊C₁₆, C₂×C₄₈, S₃×C₂×C₈, D₆⋊C₁₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, D₄, D₆, C₁₆, C₂²⋊C₄, C₂×C₈, M₄(2), C₄×S₃, D₁₂, C₃⋊D₄, C₂²⋊C₈, C₂×C₁₆, M₅(2), S₃×C₈, C₈⋊S₃, D₆⋊C₄, C₂²⋊C₁₆, S₃×C₁₆, D₆.C₈, D₆⋊C₈, D₆⋊C₁₆

Smallest permutation representation of D₆⋊C₁₆
►On 96 points

Generators in S₉₆

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	6A	6B	6C	8A	···	8H	8I	8J	8K	8L	12A	12B	12C	12D	16A	···	16H	16I	···	16P	24A	···	24H	48A	···	48P
order	1	2	2	2	2	2	3	4	4	4	4	4	4	6	6	6	8	···	8	8	8	8	8	12	12	12	12	16	···	16	16	···	16	24	···	24	48	···	48
size	1	1	1	1	6	6	2	1	1	1	1	6	6	2	2	2	1	···	1	6	6	6	6	2	2	2	2	2	···	2	6	···	6	2	···	2	2	···	2

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+						+	+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	C₈	C₁₆	S₃	D₄	D₆	M₄(2)	D₁₂	C₃⋊D₄	C₄×S₃	M₅(2)	C₈⋊S₃	S₃×C₈	S₃×C₁₆	D₆.C₈
kernel	D₆⋊C₁₆	C₂×C₃⋊C₁₆	C₂×C₄₈	S₃×C₂×C₈	C₂×C₃⋊C₈	S₃×C₂×C₄	C₂×Dic₃	C₂²×S₃	D₆	C₂×C₁₆	C₂₄	C₂×C₈	C₁₂	C₈	C₈	C₂×C₄	C₆	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	2	2	4	4	16	1	2	1	2	2	2	2	4	4	4	8	8

Matrix representation of D₆⋊C₁₆ ►in GL₄(𝔽₉₇) generated by

96	96	0	0
1	0	0	0
0	0	1	1
0	0	96	0

96	96	0	0
0	1	0	0
0	0	1	1
0	0	0	96

27	0	0	0
0	27	0	0
0	0	38	76
0	0	21	59

G:=sub<GL(4,GF(97))| [96,1,0,0,96,0,0,0,0,0,1,96,0,0,1,0],[96,0,0,0,96,1,0,0,0,0,1,0,0,0,1,96],[27,0,0,0,0,27,0,0,0,0,38,21,0,0,76,59] >;

D₆⋊C₁₆ in GAP, Magma, Sage, TeX

D_6\rtimes C_{16}

% in TeX

G:=Group("D6:C16");

// GroupNames label

G:=SmallGroup(192,66);

// by ID

G=gap.SmallGroup(192,66);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,100,102,6278]);

// Polycyclic

G:=Group<a,b,c|a^6=b^2=c^16=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;

// generators/relations

G = D6⋊C16 order 192 = 26·3

The semidirect product of D6 and C16 acting via C16/C8=C2

G = D₆⋊C₁₆ order 192 = 2⁶·3

The semidirect product of D₆ and C₁₆ acting via C₁₆/C₈=C₂