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## G = D12⋊22D4order 192 = 26·3

### 10th semidirect product of D12 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D12⋊22D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — S3×C4⋊C4 — D12⋊22D4
 Lower central C3 — C2×C6 — D12⋊22D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for D1222D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a5, cbc-1=a4b, dbd=a10b, dcd=c-1 >

Subgroups: 752 in 292 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, Dic3, Dic3, C12, C12, D6, D6, 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, C4○D4, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, 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, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, Q83S3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×Q8, D46D4, C23.9D6, Dic3⋊D4, S3×C4⋊C4, Dic35D4, D6.D4, C4.D12, C4×C3⋊D4, Dic3⋊Q8, C3×C22⋊Q8, C2×C4○D12, C2×Q83S3, D1222D4
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, D46D4, C2×S3×D4, Q8.15D6, S3×C4○D4, D1222D4

Smallest permutation representation of D1222D4
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)(2 95)(3 94)(4 93)(5 92)(6 91)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 72)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(49 75)(50 74)(51 73)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)
(1 72 45 76)(2 65 46 81)(3 70 47 74)(4 63 48 79)(5 68 37 84)(6 61 38 77)(7 66 39 82)(8 71 40 75)(9 64 41 80)(10 69 42 73)(11 62 43 78)(12 67 44 83)(13 29 49 85)(14 34 50 90)(15 27 51 95)(16 32 52 88)(17 25 53 93)(18 30 54 86)(19 35 55 91)(20 28 56 96)(21 33 57 89)(22 26 58 94)(23 31 59 87)(24 36 60 92)
(2 6)(3 11)(5 9)(8 12)(13 55)(14 60)(15 53)(16 58)(17 51)(18 56)(19 49)(20 54)(21 59)(22 52)(23 57)(24 50)(25 27)(26 32)(28 30)(29 35)(31 33)(34 36)(37 41)(38 46)(40 44)(43 47)(61 81)(62 74)(63 79)(64 84)(65 77)(66 82)(67 75)(68 80)(69 73)(70 78)(71 83)(72 76)(85 91)(86 96)(87 89)(88 94)(90 92)(93 95)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 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 4 6 6 6 6 12 2 2 2 2 2 4 4 4 4 6 6 6 6 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 D6 D6 D6 D6 C4○D4 2- 1+4 S3×D4 Q8.15D6 S3×C4○D4 kernel D12⋊22D4 C23.9D6 Dic3⋊D4 S3×C4⋊C4 Dic3⋊5D4 D6.D4 C4.D12 C4×C3⋊D4 Dic3⋊Q8 C3×C22⋊Q8 C2×C4○D12 C2×Q8⋊3S3 C22⋊Q8 D12 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 Dic3 C6 C4 C2 C2 # reps 1 2 2 2 1 2 1 1 1 1 1 1 1 4 2 3 1 1 4 1 2 2 2

Matrix representation of D1222D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 8 0 0 0 0 0 5 5
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 8 3 0 0 0 0 5 5
,
 12 10 0 0 0 0 5 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 8 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,8,5,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,8,5,0,0,0,0,3,5],[12,5,0,0,0,0,10,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,8,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

D1222D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{22}D_4
% in TeX

G:=Group("D12:22D4");
// GroupNames label

G:=SmallGroup(192,1190);
// by ID

G=gap.SmallGroup(192,1190);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,570,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=a^4*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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