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## G = C6.D16order 192 = 26·3

### 2nd non-split extension by C6 of D16 acting via D16/D8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C6.D16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C2×D24 — C6.D16
 Lower central C3 — C6 — C12 — C24 — C6.D16
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C6.D16
G = < a,b,c | a6=b16=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Subgroups: 304 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, C12, C12, D6, C2×C6, C16, C4⋊C4, C2×C8, D8, C2×D4, C24, D12, C2×C12, C2×C12, C22×S3, C2.D8, C2×C16, C2×D8, C3⋊C16, D24, D24, C3×C4⋊C4, C2×C24, C2×D12, C2.D16, C2×C3⋊C16, C3×C2.D8, C2×D24, C6.D16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D16, SD32, D6⋊C4, D4⋊S3, Q82S3, C2.D16, C6.D8, C3⋊D16, C8.6D6, C6.D16

Smallest permutation representation of C6.D16
On 96 points
Generators in S96
(1 28 83 71 34 53)(2 54 35 72 84 29)(3 30 85 73 36 55)(4 56 37 74 86 31)(5 32 87 75 38 57)(6 58 39 76 88 17)(7 18 89 77 40 59)(8 60 41 78 90 19)(9 20 91 79 42 61)(10 62 43 80 92 21)(11 22 93 65 44 63)(12 64 45 66 94 23)(13 24 95 67 46 49)(14 50 47 68 96 25)(15 26 81 69 48 51)(16 52 33 70 82 27)
(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)
(2 70)(3 15)(4 68)(5 13)(6 66)(7 11)(8 80)(10 78)(12 76)(14 74)(16 72)(17 94)(18 63)(19 92)(20 61)(21 90)(22 59)(23 88)(24 57)(25 86)(26 55)(27 84)(28 53)(29 82)(30 51)(31 96)(32 49)(33 54)(34 83)(35 52)(36 81)(37 50)(38 95)(39 64)(40 93)(41 62)(42 91)(43 60)(44 89)(45 58)(46 87)(47 56)(48 85)(65 77)(67 75)(69 73)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 16A ··· 16H 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 24 24 24 size 1 1 1 1 24 24 2 2 2 8 8 2 2 2 2 2 2 2 4 4 8 8 8 8 6 ··· 6 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 D8 C4×S3 D12 C3⋊D4 D16 SD32 Q8⋊2S3 D4⋊S3 C3⋊D16 C8.6D6 kernel C6.D16 C2×C3⋊C16 C3×C2.D8 C2×D24 D24 C2.D8 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 4 1 1 2 2

Matrix representation of C6.D16 in GL5(𝔽97)

 96 0 0 0 0 0 96 96 0 0 0 1 0 0 0 0 0 0 96 0 0 0 0 0 96
,
 75 0 0 0 0 0 1 0 0 0 0 96 96 0 0 0 0 0 54 35 0 0 0 3 63
,
 96 0 0 0 0 0 1 0 0 0 0 96 96 0 0 0 0 0 1 0 0 0 0 64 96

G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,96,1,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,96],[75,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,54,3,0,0,0,35,63],[96,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,1,64,0,0,0,0,96] >;

C6.D16 in GAP, Magma, Sage, TeX

C_6.D_{16}
% in TeX

G:=Group("C6.D16");
// GroupNames label

G:=SmallGroup(192,50);
// by ID

G=gap.SmallGroup(192,50);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,675,346,192,1684,851,102,6278]);
// Polycyclic

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

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