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## G = Dic6.D4order 192 = 26·3

### 8th non-split extension by Dic6 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic6.D4
G = < a,b,c,d | a12=c4=1, b2=d2=a6, bab-1=dad-1=a-1, cac-1=a5, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 312 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, D4.S3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C6×D4, Q8.D4, C6.SD16, Dic3⋊C8, C3×D4⋊C4, Dic6⋊C4, C2×Dic12, C2×D4.S3, C23.12D6, Dic6.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D8, C8.C22, C4○D12, S3×D4, Q8.D4, Dic3⋊D4, D83S3, D4.D6, Dic6.D4

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 8 2 2 2 4 4 6 6 12 12 12 24 2 2 2 8 8 4 4 12 12 4 4 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 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C4○D8 C4○D12 C8.C22 S3×D4 S3×D4 D8⋊3S3 D4.D6 kernel Dic6.D4 C6.SD16 Dic3⋊C8 C3×D4⋊C4 Dic6⋊C4 C2×Dic12 C2×D4.S3 C23.12D6 D4⋊C4 Dic6 C2×Dic3 C4⋊C4 C2×C8 C2×D4 C12 C6 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 1 1 1 2 2

Matrix representation of Dic6.D4 in GL4(𝔽73) generated by

 0 72 0 0 1 0 0 0 0 0 65 0 0 0 4 9
,
 67 6 0 0 6 6 0 0 0 0 56 19 0 0 54 17
,
 27 0 0 0 0 27 0 0 0 0 52 2 0 0 71 21
,
 27 0 0 0 0 46 0 0 0 0 52 2 0 0 72 21
G:=sub<GL(4,GF(73))| [0,1,0,0,72,0,0,0,0,0,65,4,0,0,0,9],[67,6,0,0,6,6,0,0,0,0,56,54,0,0,19,17],[27,0,0,0,0,27,0,0,0,0,52,71,0,0,2,21],[27,0,0,0,0,46,0,0,0,0,52,72,0,0,2,21] >;

Dic6.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6.D_4
% in TeX

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

G:=SmallGroup(192,326);
// by ID

G=gap.SmallGroup(192,326);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,1094,135,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=d^2=a^6,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^5,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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