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## G = Dic6⋊20D4order 192 = 26·3

### 8th semidirect product of Dic6 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊20D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C2×Dic6 — C2×C4○D12 — Dic6⋊20D4
 Lower central C3 — C2×C6 — Dic6⋊20D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for Dic620D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=dbd=a6b, dcd=c-1 >

Subgroups: 864 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C41D4, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, D42S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, Q86D4, Dic34D4, Dic3⋊D4, Dic6⋊C4, C12⋊D4, C4×C3⋊D4, C23.14D6, C123D4, C123D4, C3×C4⋊D4, C2×C4○D12, C2×D42S3, Dic620D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, S3×C23, Q86D4, C2×S3×D4, D46D6, S3×C4○D4, Dic620D4

Smallest permutation representation of Dic620D4
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 52 7 58)(2 51 8 57)(3 50 9 56)(4 49 10 55)(5 60 11 54)(6 59 12 53)(13 85 19 91)(14 96 20 90)(15 95 21 89)(16 94 22 88)(17 93 23 87)(18 92 24 86)(25 80 31 74)(26 79 32 73)(27 78 33 84)(28 77 34 83)(29 76 35 82)(30 75 36 81)(37 70 43 64)(38 69 44 63)(39 68 45 62)(40 67 46 61)(41 66 47 72)(42 65 48 71)
(1 40 76 92)(2 47 77 87)(3 42 78 94)(4 37 79 89)(5 44 80 96)(6 39 81 91)(7 46 82 86)(8 41 83 93)(9 48 84 88)(10 43 73 95)(11 38 74 90)(12 45 75 85)(13 53 68 36)(14 60 69 31)(15 55 70 26)(16 50 71 33)(17 57 72 28)(18 52 61 35)(19 59 62 30)(20 54 63 25)(21 49 64 32)(22 56 65 27)(23 51 66 34)(24 58 67 29)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 73)(36 74)(61 95)(62 96)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 2+ 1+4 S3×D4 D4⋊6D6 S3×C4○D4 kernel Dic6⋊20D4 Dic3⋊4D4 Dic3⋊D4 Dic6⋊C4 C12⋊D4 C4×C3⋊D4 C23.14D6 C12⋊3D4 C3×C4⋊D4 C2×C4○D12 C2×D4⋊2S3 C4⋊D4 Dic6 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 C6 C4 C2 C2 # reps 1 2 2 1 1 1 2 3 1 1 1 1 4 2 1 1 3 4 1 2 2 2

Matrix representation of Dic620D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

Dic620D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{20}D_4
% in TeX

G:=Group("Dic6:20D4");
// GroupNames label

G:=SmallGroup(192,1158);
// by ID

G=gap.SmallGroup(192,1158);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,184,570,185,6278]);
// 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=d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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