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## G = Dic6⋊8D4order 192 = 26·3

### 1st semidirect product of Dic6 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊8D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C4×Dic6 — Dic6⋊8D4
 Lower central C3 — C6 — C2×C12 — Dic6⋊8D4
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for Dic68D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a9b, dcd=c-1 >

Subgroups: 488 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C24, Dic6, Dic6, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, C24⋊C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C4⋊SD16, C2.D24, C3×C4⋊C8, C4×Dic6, C4⋊D12, C2×C24⋊C2, Dic68D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, C24⋊C2, C2×D12, S3×D4, Q83S3, C4⋊SD16, C12⋊D4, C2×C24⋊C2, C8⋊D6, Dic68D4

Smallest permutation representation of Dic68D4
On 96 points
Generators in S96
(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 20 7 14)(2 19 8 13)(3 18 9 24)(4 17 10 23)(5 16 11 22)(6 15 12 21)(25 82 31 76)(26 81 32 75)(27 80 33 74)(28 79 34 73)(29 78 35 84)(30 77 36 83)(37 93 43 87)(38 92 44 86)(39 91 45 85)(40 90 46 96)(41 89 47 95)(42 88 48 94)(49 64 55 70)(50 63 56 69)(51 62 57 68)(52 61 58 67)(53 72 59 66)(54 71 60 65)
(1 32 40 67)(2 33 41 68)(3 34 42 69)(4 35 43 70)(5 36 44 71)(6 25 45 72)(7 26 46 61)(8 27 47 62)(9 28 48 63)(10 29 37 64)(11 30 38 65)(12 31 39 66)(13 80 95 57)(14 81 96 58)(15 82 85 59)(16 83 86 60)(17 84 87 49)(18 73 88 50)(19 74 89 51)(20 75 90 52)(21 76 91 53)(22 77 92 54)(23 78 93 55)(24 79 94 56)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 36)(10 35)(11 34)(12 33)(13 73)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 72)(48 71)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)

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

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

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

39 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 12G 12H 24A ··· 24H 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 12 12 24 ··· 24 size 1 1 1 1 24 24 2 2 2 2 2 4 12 12 12 12 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C4○D4 D12 C24⋊C2 C8⋊C22 S3×D4 Q8⋊3S3 C8⋊D6 kernel Dic6⋊8D4 C2.D24 C3×C4⋊C8 C4×Dic6 C4⋊D12 C2×C24⋊C2 C4⋊C8 Dic6 C2×C12 C42 C2×C8 C12 C12 C2×C4 C4 C6 C4 C4 C2 # reps 1 2 1 1 1 2 1 2 2 1 2 4 2 4 8 1 1 1 2

Matrix representation of Dic68D4 in GL4(𝔽73) generated by

 59 66 0 0 7 66 0 0 0 0 1 0 0 0 0 1
,
 25 62 0 0 37 48 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 44 71 0 0 56 29
,
 66 59 0 0 66 7 0 0 0 0 29 2 0 0 18 44
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,1,0,0,0,0,1],[25,37,0,0,62,48,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,44,56,0,0,71,29],[66,66,0,0,59,7,0,0,0,0,29,18,0,0,2,44] >;

Dic68D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_8D_4
% in TeX

G:=Group("Dic6:8D4");
// GroupNames label

G:=SmallGroup(192,407);
// by ID

G=gap.SmallGroup(192,407);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,58,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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