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## G = Dic6⋊14D4order 192 = 26·3

### 2nd semidirect product of Dic6 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊14D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C22×Dic6 — Dic6⋊14D4
 Lower central C3 — C6 — C2×C12 — Dic6⋊14D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for Dic614D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=cac-1=a-1, ad=da, cbc-1=a9b, bd=db, dcd=c-1 >

Subgroups: 512 in 158 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic6, Dic6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C24⋊C2, C4⋊Dic3, D6⋊C4, C2×C24, C2×Dic6, C2×Dic6, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, Q8⋊D4, C2.Dic12, C3×C22⋊C8, C2×C24⋊C2, C127D4, C22×Dic6, Dic614D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C22≀C2, C2×SD16, C8.C22, C24⋊C2, C2×D12, S3×D4, Q8⋊D4, D6⋊D4, C2×C24⋊C2, C8.D6, Dic614D4

Smallest permutation representation of Dic614D4
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 37 7 43)(2 48 8 42)(3 47 9 41)(4 46 10 40)(5 45 11 39)(6 44 12 38)(13 94 19 88)(14 93 20 87)(15 92 21 86)(16 91 22 85)(17 90 23 96)(18 89 24 95)(25 54 31 60)(26 53 32 59)(27 52 33 58)(28 51 34 57)(29 50 35 56)(30 49 36 55)(61 74 67 80)(62 73 68 79)(63 84 69 78)(64 83 70 77)(65 82 71 76)(66 81 72 75)
(1 17 80 52)(2 16 81 51)(3 15 82 50)(4 14 83 49)(5 13 84 60)(6 24 73 59)(7 23 74 58)(8 22 75 57)(9 21 76 56)(10 20 77 55)(11 19 78 54)(12 18 79 53)(25 48 94 72)(26 47 95 71)(27 46 96 70)(28 45 85 69)(29 44 86 68)(30 43 87 67)(31 42 88 66)(32 41 89 65)(33 40 90 64)(34 39 91 63)(35 38 92 62)(36 37 93 61)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 61)(34 62)(35 63)(36 64)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 85)(45 86)(46 87)(47 88)(48 89)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 SD16 D12 D12 C24⋊C2 C8.C22 S3×D4 C8.D6 kernel Dic6⋊14D4 C2.Dic12 C3×C22⋊C8 C2×C24⋊C2 C12⋊7D4 C22×Dic6 C22⋊C8 Dic6 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 C6 C4 C2 # reps 1 2 1 2 1 1 1 4 1 1 2 1 4 2 2 8 1 2 2

Matrix representation of Dic614D4 in GL6(𝔽73)

 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 67 6 0 0 0 0 6 6 0 0 0 0 0 0 0 72 0 0 0 0 72 0
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 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 72 0

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

Dic614D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{14}D_4
% in TeX

G:=Group("Dic6:14D4");
// GroupNames label

G:=SmallGroup(192,297);
// by ID

G=gap.SmallGroup(192,297);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,58,1123,136,6278]);
// Polycyclic

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

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