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G = Dic6:20D4order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic6:20D4, C6.332+ 1+4, C4:D4:7S3, C4:C4.177D6, (C2xD4).90D6, C3:3(Q8:6D4), C4.109(S3xD4), Dic3:D4:17C2, C12:D4:19C2, C12:3D4:15C2, C12.225(C2xD4), C22:C4.46D6, C6.62(C22xD4), Dic3:8(C4oD4), Dic3:4D4:6C2, (C2xC12).35C23, (C2xC6).143C24, D6:C4.12C22, Dic3.21(C2xD4), (C22xC4).235D6, Dic6:C4:20C2, C23.14D6:10C2, C2.35(D4:6D6), (C6xD4).117C22, C23.20(C22xS3), (C22xC6).14C23, (C2xD12).142C22, Dic3:C4.14C22, (C22xS3).62C23, C22.164(S3xC23), (C4xDic3).90C22, (C22xC12).237C22, (C2xDic6).293C22, (C2xDic3).225C23, C6.D4.110C22, (C22xDic3).104C22, C2.35(C2xS3xD4), (C3xC4:D4):8C2, (C4xC3:D4):15C2, C2.34(S3xC4oD4), (C2xC4oD12):19C2, C6.148(C2xC4oD4), (S3xC2xC4).82C22, (C2xD4:2S3):11C2, (C3xC4:C4).139C22, (C2xC4).585(C22xS3), (C2xC3:D4).25C22, (C3xC22:C4).8C22, SmallGroup(192,1158)

Series: Derived Chief Lower central Upper central

C1C2xC6 — Dic6:20D4
C1C3C6C2xC6C2xDic3C2xDic6C2xC4oD12 — Dic6:20D4
C3C2xC6 — Dic6:20D4
C1C22C4:D4

Generators and relations for Dic6:20D4
 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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22xC6, C4xD4, C4xQ8, C4:D4, C4:D4, C4:1D4, C2xC4oD4, C4xDic3, C4xDic3, Dic3:C4, Dic3:C4, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C4oD12, D4:2S3, C22xDic3, C2xC3:D4, C2xC3:D4, C22xC12, C6xD4, C6xD4, Q8:6D4, Dic3:4D4, Dic3:D4, Dic6:C4, C12:D4, C4xC3:D4, C23.14D6, C12:3D4, C12:3D4, C3xC4:D4, C2xC4oD12, C2xD4:2S3, Dic6:20D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2+ 1+4, S3xD4, S3xC23, Q8:6D4, C2xS3xD4, D4:6D6, S3xC4oD4, Dic6:20D4

Smallest permutation representation of Dic6:20D4
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 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G···4N4O6A6B6C6D6E6F6G12A12B12C12D12E12F
order122222222234444444···446666666121212121212
size111144412121222222446···6122224488444488

39 irreducible representations

dim1111111111122222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4oD42+ 1+4S3xD4D4:6D6S3xC4oD4
kernelDic6:20D4Dic3:4D4Dic3:D4Dic6:C4C12:D4C4xC3:D4C23.14D6C12:3D4C3xC4:D4C2xC4oD12C2xD4:2S3C4:D4Dic6C22:C4C4:C4C22xC4C2xD4Dic3C6C4C2C2
# reps1221112311114211341222

Matrix representation of Dic6:20D4 in GL6(F13)

100000
010000
000100
0012000
0000121
0000120
,
1200000
0120000
008000
000500
0000012
0000120
,
010000
1200000
000500
005000
000010
000001
,
1200000
010000
000500
008000
0000120
0000012

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] >;

Dic6:20D4 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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