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## G = D6⋊C16order 192 = 26·3

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

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

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