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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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C2×C3⋊C8, C3⋊Q16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×Q8, C22⋊Q16, C6.SD16, C12.55D4, C2×C3⋊Q16, C3×C22⋊Q8, C22×Dic6, Dic6.37D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×Q16, C8.C22, C3⋊Q16, S3×D4, 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 75 7 81)(2 74 8 80)(3 73 9 79)(4 84 10 78)(5 83 11 77)(6 82 12 76)(13 34 19 28)(14 33 20 27)(15 32 21 26)(16 31 22 25)(17 30 23 36)(18 29 24 35)(37 55 43 49)(38 54 44 60)(39 53 45 59)(40 52 46 58)(41 51 47 57)(42 50 48 56)(61 91 67 85)(62 90 68 96)(63 89 69 95)(64 88 70 94)(65 87 71 93)(66 86 72 92)
(1 31 90 38)(2 26 91 45)(3 33 92 40)(4 28 93 47)(5 35 94 42)(6 30 95 37)(7 25 96 44)(8 32 85 39)(9 27 86 46)(10 34 87 41)(11 29 88 48)(12 36 89 43)(13 68 57 75)(14 63 58 82)(15 70 59 77)(16 65 60 84)(17 72 49 79)(18 67 50 74)(19 62 51 81)(20 69 52 76)(21 64 53 83)(22 71 54 78)(23 66 55 73)(24 61 56 80)
(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)

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

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

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

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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