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## G = D4.3Dic6order 192 = 26·3

### The non-split extension by D4 of Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D4.3Dic6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C12.6Q8 — D4.3Dic6
 Lower central C3 — C6 — C2×C12 — D4.3Dic6
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.3Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=a2c6, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 248 in 102 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C2×D4, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×C6, D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, D4.Q8, C12⋊C8, C6.Q16, C12.Q8, D4⋊Dic3 [×2], C12.6Q8, D4×C12, D4.3Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C2×Dic6, C4○D12, C2×C3⋊D4, D4.Q8, C12.48D4, D126C22, Q8.13D6, D4.3Dic6

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D8 C4○D12 C8⋊C22 D12⋊6C22 Q8.13D6 kernel D4.3Dic6 C12⋊C8 C6.Q16 C12.Q8 D4⋊Dic3 C12.6Q8 D4×C12 C4×D4 C2×C12 C3×D4 C42 C4⋊C4 C2×D4 C12 C2×C4 D4 C6 C4 C6 C2 C2 # reps 1 1 1 1 2 1 1 1 2 2 1 1 1 2 4 4 4 4 1 2 2

Matrix representation of D4.3Dic6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 71 0 0 0 0 1 72
,
 72 0 0 0 0 0 40 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 71 0 0 0 0 0 72
,
 9 0 0 0 0 0 25 65 0 0 0 0 0 0 46 0 0 0 0 0 27 27 0 0 0 0 0 0 46 0 0 0 0 0 0 46
,
 40 2 0 0 0 0 40 33 0 0 0 0 0 0 72 71 0 0 0 0 1 1 0 0 0 0 0 0 0 32 0 0 0 0 16 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[72,40,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72],[9,25,0,0,0,0,0,65,0,0,0,0,0,0,46,27,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[40,40,0,0,0,0,2,33,0,0,0,0,0,0,72,1,0,0,0,0,71,1,0,0,0,0,0,0,0,16,0,0,0,0,32,0] >;

D4.3Dic6 in GAP, Magma, Sage, TeX

D_4._3{\rm Dic}_6
% in TeX

G:=Group("D4.3Dic6");
// GroupNames label

G:=SmallGroup(192,568);
// by ID

G=gap.SmallGroup(192,568);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,1123,297,136,6278]);
// Polycyclic

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

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