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## G = Dic6.37D4order 192 = 26·3

### 7th non-split extension by Dic6 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6.37D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C22×Dic6 — Dic6.37D4
 Lower central C3 — C6 — C2×C12 — Dic6.37D4
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for Dic6.37D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, bd=db, dcd=a6c-1 >

Subgroups: 384 in 148 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×12], C23, Dic3 [×4], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8, C2×Q8 [×7], C3⋊C8 [×2], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C2×C3⋊C8 [×2], C3⋊Q16 [×4], C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6 [×2], C2×Dic6 [×5], C22×Dic3, C22×C12, C6×Q8, C22⋊Q16, C6.SD16 [×2], C12.55D4, C2×C3⋊Q16 [×2], C3×C22⋊Q8, C22×Dic6, Dic6.37D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], Q16 [×2], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C2×Q16, C8.C22, C3⋊Q16 [×2], S3×D4 [×2], C2×C3⋊D4, C22⋊Q16, C232D6, C2×C3⋊Q16, Q8.14D6, Dic6.37D4

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - + - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 Q16 C3⋊D4 C3⋊D4 C8.C22 S3×D4 C3⋊Q16 Q8.14D6 kernel Dic6.37D4 C6.SD16 C12.55D4 C2×C3⋊Q16 C3×C22⋊Q8 C22×Dic6 C22⋊Q8 Dic6 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×Q8 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 4 1 1 1 1 1 4 2 2 1 2 2 2

Matrix representation of Dic6.37D4 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 11 60 0 0 0 0 71 62 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 60 66 0 0 0 0 66 13
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 56 70 0 0 0 0 48 17 0 0 0 0 0 0 50 45 0 0 0 0 45 23
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 13 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[11,71,0,0,0,0,60,62,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,60,66,0,0,0,0,66,13],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,56,48,0,0,0,0,70,17,0,0,0,0,0,0,50,45,0,0,0,0,45,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic6.37D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{37}D_4
% in TeX

G:=Group("Dic6.37D4");
// GroupNames label

G:=SmallGroup(192,609);
// by ID

G=gap.SmallGroup(192,609);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d=a^6*c^-1>;
// generators/relations

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