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## G = D6.2SD16order 192 = 26·3

### 2nd non-split extension by D6 of SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D6.2SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — S3×C4⋊C4 — D6.2SD16
 Lower central C3 — C6 — C2×C12 — D6.2SD16
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for D6.2SD16
G = < a,b,c,d | a6=b2=c8=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c3 >

Subgroups: 304 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊C8, Q8⋊C4, C4.Q8, C4.Q8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, S3×C2×C4, C23.47D4, C12.Q8, C6.SD16, C2.Dic12, D6⋊C8, C3×C4.Q8, S3×C4⋊C4, C4.D12, D6.2SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C22×S3, C22.D4, C2×SD16, C8.C22, C4○D12, S3×D4, Q83S3, C23.47D4, D6.D4, S3×SD16, D4.D6, D6.2SD16

Smallest permutation representation of D6.2SD16
On 96 points
Generators in S96
(1 45 22 77 53 62)(2 46 23 78 54 63)(3 47 24 79 55 64)(4 48 17 80 56 57)(5 41 18 73 49 58)(6 42 19 74 50 59)(7 43 20 75 51 60)(8 44 21 76 52 61)(9 29 39 95 83 69)(10 30 40 96 84 70)(11 31 33 89 85 71)(12 32 34 90 86 72)(13 25 35 91 87 65)(14 26 36 92 88 66)(15 27 37 93 81 67)(16 28 38 94 82 68)
(1 58)(2 19)(3 60)(4 21)(5 62)(6 23)(7 64)(8 17)(9 65)(10 36)(11 67)(12 38)(13 69)(14 40)(15 71)(16 34)(18 77)(20 79)(22 73)(24 75)(25 83)(26 30)(27 85)(28 32)(29 87)(31 81)(33 93)(35 95)(37 89)(39 91)(41 53)(42 46)(43 55)(44 48)(45 49)(47 51)(50 54)(52 56)(57 76)(59 78)(61 80)(63 74)(66 96)(68 90)(70 92)(72 94)(82 86)(84 88)
(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)
(1 9 77 95)(2 12 78 90)(3 15 79 93)(4 10 80 96)(5 13 73 91)(6 16 74 94)(7 11 75 89)(8 14 76 92)(17 40 57 70)(18 35 58 65)(19 38 59 68)(20 33 60 71)(21 36 61 66)(22 39 62 69)(23 34 63 72)(24 37 64 67)(25 49 87 41)(26 52 88 44)(27 55 81 47)(28 50 82 42)(29 53 83 45)(30 56 84 48)(31 51 85 43)(32 54 86 46)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 SD16 C4○D12 C8.C22 Q8⋊3S3 S3×D4 S3×SD16 D4.D6 kernel D6.2SD16 C12.Q8 C6.SD16 C2.Dic12 D6⋊C8 C3×C4.Q8 S3×C4⋊C4 C4.D12 C4.Q8 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C12 D6 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 4 4 4 1 1 1 2 2

Matrix representation of D6.2SD16 in GL4(𝔽73) generated by

 1 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 1 1 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 30 60 0 0 13 43 0 0 0 0 67 6 0 0 67 67
,
 46 0 0 0 0 46 0 0 0 0 67 44 0 0 44 6
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,0,72],[30,13,0,0,60,43,0,0,0,0,67,67,0,0,6,67],[46,0,0,0,0,46,0,0,0,0,67,44,0,0,44,6] >;

D6.2SD16 in GAP, Magma, Sage, TeX

D_6._2{\rm SD}_{16}
% in TeX

G:=Group("D6.2SD16");
// GroupNames label

G:=SmallGroup(192,421);
// by ID

G=gap.SmallGroup(192,421);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,254,219,100,851,102,6278]);
// Polycyclic

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

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