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G = Dic6:22D4order 192 = 26·3

10th semidirect product of Dic6 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic6:22D4, C6.192- 1+4, C4:C4.191D6, C3:4(Q8:5D4), C4.114(S3xD4), C22:Q8:10S3, D6:14(C4oD4), Dic3:D4:26C2, C12:D4:26C2, D6:Q8:20C2, C12.237(C2xD4), (C2xQ8).152D6, C22:C4.17D6, Dic3:5D4:27C2, C6.79(C22xD4), (C2xC12).56C23, (C2xC6).177C24, Dic3.25(C2xD4), (C22xC4).255D6, Dic6:C4:26C2, C12.23D4:13C2, D6:C4.128C22, (C6xQ8).109C22, C23.11D6:25C2, (C2xD12).265C22, Dic3:C4.29C22, C23.130(C22xS3), C22.198(S3xC23), (C22xC6).205C23, (C22xS3).199C23, (C22xC12).257C22, C2.20(Q8.15D6), (C2xDic6).295C22, (C4xDic3).107C22, (C2xDic3).236C23, C6.D4.118C22, (C2xS3xQ8):8C2, C2.52(C2xS3xD4), (C4xC3:D4):24C2, C2.50(S3xC4oD4), (C2xC4oD12):25C2, C6.162(C2xC4oD4), (S3xC2xC4).97C22, (C3xC22:Q8):13C2, (C3xC4:C4).160C22, (C2xC4).592(C22xS3), (C2xC3:D4).125C22, (C3xC22:C4).32C22, SmallGroup(192,1192)

Series: Derived Chief Lower central Upper central

C1C2xC6 — Dic6:22D4
C1C3C6C2xC6C22xS3S3xC2xC4C2xS3xQ8 — Dic6:22D4
C3C2xC6 — Dic6:22D4
C1C22C22:Q8

Generators and relations for Dic6:22D4
 G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=dad=a5, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 752 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C2xQ8, C4oD4, Dic6, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C22xC6, C4xD4, C4xQ8, C4:D4, C22:Q8, C22:Q8, C4.4D4, C22xQ8, C2xC4oD4, C4xDic3, C4xDic3, Dic3:C4, Dic3:C4, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C4oD12, S3xQ8, C2xC3:D4, C2xC3:D4, C22xC12, C6xQ8, Q8:5D4, Dic3:D4, C23.11D6, Dic6:C4, Dic3:5D4, C12:D4, D6:Q8, C4xC3:D4, C12.23D4, C3xC22:Q8, C2xC4oD12, C2xS3xQ8, Dic6:22D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2- 1+4, S3xD4, S3xC23, Q8:5D4, C2xS3xD4, Q8.15D6, S3xC4oD4, Dic6:22D4

Smallest permutation representation of Dic6:22D4
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 69 7 63)(2 68 8 62)(3 67 9 61)(4 66 10 72)(5 65 11 71)(6 64 12 70)(13 35 19 29)(14 34 20 28)(15 33 21 27)(16 32 22 26)(17 31 23 25)(18 30 24 36)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)(49 96 55 90)(50 95 56 89)(51 94 57 88)(52 93 58 87)(53 92 59 86)(54 91 60 85)
(1 95 77 26)(2 88 78 31)(3 93 79 36)(4 86 80 29)(5 91 81 34)(6 96 82 27)(7 89 83 32)(8 94 84 25)(9 87 73 30)(10 92 74 35)(11 85 75 28)(12 90 76 33)(13 72 53 43)(14 65 54 48)(15 70 55 41)(16 63 56 46)(17 68 57 39)(18 61 58 44)(19 66 59 37)(20 71 60 42)(21 64 49 47)(22 69 50 40)(23 62 51 45)(24 67 52 38)
(1 22)(2 15)(3 20)(4 13)(5 18)(6 23)(7 16)(8 21)(9 14)(10 19)(11 24)(12 17)(25 64)(26 69)(27 62)(28 67)(29 72)(30 65)(31 70)(32 63)(33 68)(34 61)(35 66)(36 71)(37 92)(38 85)(39 90)(40 95)(41 88)(42 93)(43 86)(44 91)(45 96)(46 89)(47 94)(48 87)(49 84)(50 77)(51 82)(52 75)(53 80)(54 73)(55 78)(56 83)(57 76)(58 81)(59 74)(60 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), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,95,77,26)(2,88,78,31)(3,93,79,36)(4,86,80,29)(5,91,81,34)(6,96,82,27)(7,89,83,32)(8,94,84,25)(9,87,73,30)(10,92,74,35)(11,85,75,28)(12,90,76,33)(13,72,53,43)(14,65,54,48)(15,70,55,41)(16,63,56,46)(17,68,57,39)(18,61,58,44)(19,66,59,37)(20,71,60,42)(21,64,49,47)(22,69,50,40)(23,62,51,45)(24,67,52,38), (1,22)(2,15)(3,20)(4,13)(5,18)(6,23)(7,16)(8,21)(9,14)(10,19)(11,24)(12,17)(25,64)(26,69)(27,62)(28,67)(29,72)(30,65)(31,70)(32,63)(33,68)(34,61)(35,66)(36,71)(37,92)(38,85)(39,90)(40,95)(41,88)(42,93)(43,86)(44,91)(45,96)(46,89)(47,94)(48,87)(49,84)(50,77)(51,82)(52,75)(53,80)(54,73)(55,78)(56,83)(57,76)(58,81)(59,74)(60,79)>;

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H4I···4N4O4P6A6B6C6D6E12A12B12C12D12E12F12G12H
order1222222223444444444···444666661212121212121212
size111146612122222244446···612122224444448888

39 irreducible representations

dim11111111111122222224444
type++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4oD42- 1+4S3xD4Q8.15D6S3xC4oD4
kernelDic6:22D4Dic3:D4C23.11D6Dic6:C4Dic3:5D4C12:D4D6:Q8C4xC3:D4C12.23D4C3xC22:Q8C2xC4oD12C2xS3xQ8C22:Q8Dic6C22:C4C4:C4C22xC4C2xQ8D6C6C4C2C2
# reps12212121111114231141222

Matrix representation of Dic6:22D4 in GL6(F13)

080000
800000
0012100
0012000
000010
000001
,
0120000
100000
000100
001000
000010
000001
,
0120000
1200000
000100
001000
000001
0000120
,
100000
010000
000100
001000
000001
000010

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

Dic6:22D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{22}D_4
% in TeX

G:=Group("Dic6:22D4");
// GroupNames label

G:=SmallGroup(192,1192);
// by ID

G=gap.SmallGroup(192,1192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,6278]);
// 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=d*a*d=a^5,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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