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G = D12.4Q8order 192 = 26·3

2nd non-split extension by D12 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.4Q8, C42.69D6, C4:C4.75D6, C3:7(D4.Q8), C4.10(S3xQ8), C12:C8:30C2, C12.34(C2xQ8), C42.C2:1S3, (C4xD12).16C2, (C2xC12).275D4, C6.Q16:42C2, C12.70(C4oD4), C6.109(C4oD8), C12.Q8:41C2, C6.D8.12C2, C6.74(C22:Q8), C2.21(D4:D6), C6.122(C8:C22), (C2xC12).384C23, (C4xC12).114C22, C2.11(D6:3Q8), C4.33(Q8:3S3), C2.28(Q8.13D6), (C2xD12).245C22, C4:Dic3.343C22, (C2xC6).515(C2xD4), (C3xC42.C2):1C2, (C2xC4).66(C3:D4), (C2xC3:C8).126C22, (C3xC4:C4).122C22, (C2xC4).482(C22xS3), C22.188(C2xC3:D4), SmallGroup(192,625)

Series: Derived Chief Lower central Upper central

C1C2xC12 — D12.4Q8
C1C3C6C12C2xC12C2xD12C4xD12 — D12.4Q8
C3C6C2xC12 — D12.4Q8
C1C22C42C42.C2

Generators and relations for D12.4Q8
 G = < a,b,c,d | a12=b2=c4=1, d2=a6c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 304 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C22xC4, C2xD4, C3:C8, C4xS3, D12, D12, C2xDic3, C2xC12, C2xC12, C22xS3, D4:C4, C4:C8, C4.Q8, C2.D8, C4xD4, C42.C2, C2xC3:C8, C4:Dic3, D6:C4, C4xC12, C3xC4:C4, C3xC4:C4, S3xC2xC4, C2xD12, D4.Q8, C12:C8, C6.Q16, C12.Q8, C6.D8, C4xD12, C3xC42.C2, D12.4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, C3:D4, C22xS3, C22:Q8, C4oD8, C8:C22, S3xQ8, Q8:3S3, C2xC3:D4, D4.Q8, D6:3Q8, D4:D6, Q8.13D6, D12.4Q8

Smallest permutation representation of D12.4Q8
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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 96)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(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 17 90 46)(2 24 91 41)(3 19 92 48)(4 14 93 43)(5 21 94 38)(6 16 95 45)(7 23 96 40)(8 18 85 47)(9 13 86 42)(10 20 87 37)(11 15 88 44)(12 22 89 39)(25 56 72 77)(26 51 61 84)(27 58 62 79)(28 53 63 74)(29 60 64 81)(30 55 65 76)(31 50 66 83)(32 57 67 78)(33 52 68 73)(34 59 69 80)(35 54 70 75)(36 49 71 82)
(1 82 96 55)(2 83 85 56)(3 84 86 57)(4 73 87 58)(5 74 88 59)(6 75 89 60)(7 76 90 49)(8 77 91 50)(9 78 92 51)(10 79 93 52)(11 80 94 53)(12 81 95 54)(13 67 48 26)(14 68 37 27)(15 69 38 28)(16 70 39 29)(17 71 40 30)(18 72 41 31)(19 61 42 32)(20 62 43 33)(21 63 44 34)(22 64 45 35)(23 65 46 36)(24 66 47 25)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D12A···12F12G12H12I12J
order1222223444444444666888812···1212121212
size11111212222224881212222121212124···48888

33 irreducible representations

dim11111112222222244444
type++++++++-++++-++
imageC1C2C2C2C2C2C2S3Q8D4D6D6C4oD4C3:D4C4oD8C8:C22S3xQ8Q8:3S3D4:D6Q8.13D6
kernelD12.4Q8C12:C8C6.Q16C12.Q8C6.D8C4xD12C3xC42.C2C42.C2D12C2xC12C42C4:C4C12C2xC4C6C6C4C4C2C2
# reps11112111221224411122

Matrix representation of D12.4Q8 in GL6(F73)

7200000
0720000
0007200
0017200
00007271
000011
,
100000
24720000
0017200
0007200
000012
0000072
,
3730000
30360000
001000
000100
0000032
0000160
,
2700000
64460000
0072000
0007200
0000460
0000046

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,71,1],[1,24,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,2,72],[37,30,0,0,0,0,3,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,32,0],[27,64,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46] >;

D12.4Q8 in GAP, Magma, Sage, TeX

D_{12}._4Q_8
% in TeX

G:=Group("D12.4Q8");
// GroupNames label

G:=SmallGroup(192,625);
// by ID

G=gap.SmallGroup(192,625);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6*c^2,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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