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G = Dic6:21D4order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic6:21D4, C6.762- 1+4, C3:3(D4xQ8), D6:6(C2xQ8), C3:D4:1Q8, C12:Q8:25C2, C22:Q8:9S3, C22:2(S3xQ8), C4:C4.190D6, Dic3:4(C2xQ8), C4.113(S3xD4), D6:Q8:19C2, C4.D12:26C2, C12.236(C2xD4), (C2xQ8).151D6, C22:C4.58D6, C6.78(C22xD4), C6.35(C22xQ8), (C2xC6).176C24, (C2xC12).55C23, C2.36(Q8oD12), Dic3.24(C2xD4), (C22xC4).254D6, Dic6:C4:25C2, Dic3:Q8:15C2, D6:C4.107C22, Dic3:4D4.1C2, (C22xDic6):17C2, (C6xQ8).108C22, Dic3.D4:23C2, Dic3:C4.28C22, C4:Dic3.216C22, (C22xC6).204C23, C22.197(S3xC23), C23.200(C22xS3), (C22xS3).198C23, (C22xC12).256C22, (C2xDic3).235C23, (C2xDic6).248C22, (C4xDic3).106C22, C6.D4.117C22, (C22xDic3).118C22, (C2xS3xQ8):7C2, (C2xC6):3(C2xQ8), C2.51(C2xS3xD4), C2.18(C2xS3xQ8), (C4xC3:D4).7C2, (S3xC2xC4).96C22, (C3xC22:Q8):12C2, (C2xC4).49(C22xS3), (C3xC4:C4).159C22, (C2xC3:D4).124C22, (C3xC22:C4).31C22, SmallGroup(192,1191)

Series: Derived Chief Lower central Upper central

C1C2xC6 — Dic6:21D4
C1C3C6C2xC6C22xS3C2xC3:D4C4xC3:D4 — Dic6:21D4
C3C2xC6 — Dic6:21D4
C1C22C22:Q8

Generators and relations for Dic6:21D4
 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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C2xQ8, Dic6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xC6, C4xD4, C4xQ8, C22:Q8, C22:Q8, C4:Q8, C22xQ8, C4xDic3, C4xDic3, Dic3:C4, Dic3:C4, C4:Dic3, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xDic6, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, S3xQ8, C22xDic3, C2xC3:D4, C22xC12, C6xQ8, D4xQ8, Dic3.D4, Dic3:4D4, Dic6:C4, C12:Q8, D6:Q8, C4.D12, C4xC3:D4, Dic3:Q8, C3xC22:Q8, C22xDic6, C2xS3xQ8, Dic6:21D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C24, C22xS3, C22xD4, C22xQ8, 2- 1+4, S3xD4, S3xQ8, S3xC23, D4xQ8, C2xS3xD4, C2xS3xQ8, Q8oD12, Dic6:21D4

Smallest permutation representation of Dic6:21D4
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 2A2B2C2D2E2F2G 3 4A4B4C···4G4H···4M4N4O4P4Q6A6B6C6D6E12A12B12C12D12E12F12G12H
order122222223444···44···44444666661212121212121212
size111122662224···46···6121212122224444448888

39 irreducible representations

dim11111111111122222224444
type++++++++++++++-++++-+--
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4Q8D6D6D6D62- 1+4S3xD4S3xQ8Q8oD12
kernelDic6:21D4Dic3.D4Dic3:4D4Dic6:C4C12:Q8D6:Q8C4.D12C4xC3:D4Dic3:Q8C3xC22:Q8C22xDic6C2xS3xQ8C22:Q8Dic6C3:D4C22:C4C4:C4C22xC4C2xQ8C6C4C22C2
# reps12212211111114423111222

Matrix representation of Dic6:21D4 in GL6(F13)

390000
9100000
0011200
001000
000010
000001
,
010000
1200000
0001200
0012000
000010
000001
,
1200000
0120000
0001200
0012000
00001211
000011
,
1200000
0120000
0012000
0001200
00001211
000001

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] >;

Dic6:21D4 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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