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## G = Dic6⋊21D4order 192 = 26·3

### 9th semidirect product of Dic6 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊21D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C4×C3⋊D4 — Dic6⋊21D4
 Lower central C3 — C2×C6 — Dic6⋊21D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for Dic621D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a5, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 656 in 280 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, D4×Q8, Dic3.D4, Dic34D4, Dic6⋊C4, C12⋊Q8, D6⋊Q8, C4.D12, C4×C3⋊D4, Dic3⋊Q8, C3×C22⋊Q8, C22×Dic6, C2×S3×Q8, Dic621D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, C22×S3, C22×D4, C22×Q8, 2- 1+4, S3×D4, S3×Q8, S3×C23, D4×Q8, C2×S3×D4, C2×S3×Q8, Q8○D12, Dic621D4

Smallest permutation representation of Dic621D4
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 56 7 50)(2 55 8 49)(3 54 9 60)(4 53 10 59)(5 52 11 58)(6 51 12 57)(13 74 19 80)(14 73 20 79)(15 84 21 78)(16 83 22 77)(17 82 23 76)(18 81 24 75)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)(61 87 67 93)(62 86 68 92)(63 85 69 91)(64 96 70 90)(65 95 71 89)(66 94 72 88)
(1 21 95 41)(2 14 96 46)(3 19 85 39)(4 24 86 44)(5 17 87 37)(6 22 88 42)(7 15 89 47)(8 20 90 40)(9 13 91 45)(10 18 92 38)(11 23 93 43)(12 16 94 48)(25 50 84 65)(26 55 73 70)(27 60 74 63)(28 53 75 68)(29 58 76 61)(30 51 77 66)(31 56 78 71)(32 49 79 64)(33 54 80 69)(34 59 81 62)(35 52 82 67)(36 57 83 72)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 85)(24 86)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 49)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 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 C2 S3 D4 Q8 D6 D6 D6 D6 2- 1+4 S3×D4 S3×Q8 Q8○D12 kernel Dic6⋊21D4 Dic3.D4 Dic3⋊4D4 Dic6⋊C4 C12⋊Q8 D6⋊Q8 C4.D12 C4×C3⋊D4 Dic3⋊Q8 C3×C22⋊Q8 C22×Dic6 C2×S3×Q8 C22⋊Q8 Dic6 C3⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C6 C4 C22 C2 # reps 1 2 2 1 2 2 1 1 1 1 1 1 1 4 4 2 3 1 1 1 2 2 2

Matrix representation of Dic621D4 in GL6(𝔽13)

 3 9 0 0 0 0 9 10 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 11 0 0 0 0 1 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 11 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [3,9,0,0,0,0,9,10,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,11,1] >;

Dic621D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{21}D_4
% in TeX

G:=Group("Dic6:21D4");
// GroupNames label

G:=SmallGroup(192,1191);
// by ID

G=gap.SmallGroup(192,1191);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,570,185,80,6278]);
// 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^5,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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