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## G = D4×Dic6order 192 = 26·3

### Direct product of D4 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D4×Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — D4×Dic3 — D4×Dic6
 Lower central C3 — C2×C6 — D4×Dic6
 Upper central C1 — C22 — C4×D4

Generators and relations for D4×Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 632 in 280 conjugacy classes, 123 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, D4×Q8, C4×Dic6, C122Q8, Dic3.D4, C12⋊Q8, C12.48D4, D4×Dic3, D4×C12, C22×Dic6, D4×Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, Dic6, C22×S3, C22×D4, C22×Q8, 2- 1+4, C2×Dic6, S3×D4, S3×C23, D4×Q8, C22×Dic6, C2×S3×D4, Q8○D12, D4×Dic6

Smallest permutation representation of D4×Dic6
On 96 points
Generators in S96
(1 65 89 28)(2 66 90 29)(3 67 91 30)(4 68 92 31)(5 69 93 32)(6 70 94 33)(7 71 95 34)(8 72 96 35)(9 61 85 36)(10 62 86 25)(11 63 87 26)(12 64 88 27)(13 60 77 37)(14 49 78 38)(15 50 79 39)(16 51 80 40)(17 52 81 41)(18 53 82 42)(19 54 83 43)(20 55 84 44)(21 56 73 45)(22 57 74 46)(23 58 75 47)(24 59 76 48)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 84)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(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 22 7 16)(2 21 8 15)(3 20 9 14)(4 19 10 13)(5 18 11 24)(6 17 12 23)(25 37 31 43)(26 48 32 42)(27 47 33 41)(28 46 34 40)(29 45 35 39)(30 44 36 38)(49 67 55 61)(50 66 56 72)(51 65 57 71)(52 64 58 70)(53 63 59 69)(54 62 60 68)(73 96 79 90)(74 95 80 89)(75 94 81 88)(76 93 82 87)(77 92 83 86)(78 91 84 85)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L ··· 4Q 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 6 6 6 6 12 ··· 12 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 D6 D6 Dic6 2- 1+4 S3×D4 Q8○D12 kernel D4×Dic6 C4×Dic6 C12⋊2Q8 Dic3.D4 C12⋊Q8 C12.48D4 D4×Dic3 D4×C12 C22×Dic6 C4×D4 Dic6 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C6 C4 C2 # reps 1 1 1 4 2 2 2 1 2 1 4 4 1 2 1 2 1 8 1 2 2

Matrix representation of D4×Dic6 in GL6(𝔽13)

 1 3 0 0 0 0 8 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 3 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 6 0 0 0 0 4 1 0 0 0 0 0 0 6 10 0 0 0 0 3 7

G:=sub<GL(6,GF(13))| [1,8,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,0,6,1,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;

D4×Dic6 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_6
% in TeX

G:=Group("D4xDic6");
// GroupNames label

G:=SmallGroup(192,1096);
// by ID

G=gap.SmallGroup(192,1096);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,675,80,6278]);
// Polycyclic

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

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