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G = D12.12D4order 192 = 26·3

12nd non-split extension by D12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.12D4, C4.97(S3×D4), (C2×C8).20D6, Dic3⋊C88C2, C4⋊C4.157D6, Q8⋊C48S3, (C2×D24).3C2, (C2×Q8).48D6, Dic35D45C2, C6.72(C4○D8), C4.8(C4○D12), C12.129(C2×D4), C6.D813C2, C32(D4.2D4), C12.24(C4○D4), C12.23D42C2, C6.28(C4⋊D4), C2.19(Q83D6), C6.66(C8⋊C22), (C2×C24).20C22, (C2×Dic3).36D4, C22.208(S3×D4), (C6×Q8).42C22, C2.31(Dic3⋊D4), (C2×C12).259C23, (C2×D12).69C22, C2.11(D24⋊C2), (C4×Dic3).26C22, (C3×Q8⋊C4)⋊8C2, (C2×Q82S3)⋊6C2, (C2×C6).272(C2×D4), (C2×C3⋊C8).49C22, (C3×C4⋊C4).60C22, (C2×C4).366(C22×S3), SmallGroup(192,378)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D12.12D4
C1C3C6C2×C6C2×C12C2×D12Dic35D4 — D12.12D4
C3C6C2×C12 — D12.12D4
C1C22C2×C4Q8⋊C4

Generators and relations for D12.12D4
 G = < a,b,c,d | a12=b2=c4=1, d2=a3, bab=a-1, cac-1=a7, ad=da, bc=cb, dbd-1=a3b, dcd-1=a9c-1 >

Subgroups: 440 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, D24, C2×C3⋊C8, C4×Dic3, D6⋊C4, Q82S3, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C6×Q8, D4.2D4, C6.D8, Dic3⋊C8, C3×Q8⋊C4, Dic35D4, C2×D24, C2×Q82S3, C12.23D4, D12.12D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D8, C8⋊C22, C4○D12, S3×D4, D4.2D4, Dic3⋊D4, Q83D6, D24⋊C2, D12.12D4

Smallest permutation representation of D12.12D4
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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 24)(9 23)(10 22)(11 21)(12 20)(25 40)(26 39)(27 38)(28 37)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 84)(59 83)(60 82)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 96)(71 95)(72 94)
(1 37 93 51)(2 44 94 58)(3 39 95 53)(4 46 96 60)(5 41 85 55)(6 48 86 50)(7 43 87 57)(8 38 88 52)(9 45 89 59)(10 40 90 54)(11 47 91 49)(12 42 92 56)(13 34 67 73)(14 29 68 80)(15 36 69 75)(16 31 70 82)(17 26 71 77)(18 33 72 84)(19 28 61 79)(20 35 62 74)(21 30 63 81)(22 25 64 76)(23 32 65 83)(24 27 66 78)
(1 68 4 71 7 62 10 65)(2 69 5 72 8 63 11 66)(3 70 6 61 9 64 12 67)(13 95 16 86 19 89 22 92)(14 96 17 87 20 90 23 93)(15 85 18 88 21 91 24 94)(25 39 28 42 31 45 34 48)(26 40 29 43 32 46 35 37)(27 41 30 44 33 47 36 38)(49 75 52 78 55 81 58 84)(50 76 53 79 56 82 59 73)(51 77 54 80 57 83 60 74)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H6A6B6C8A8B8C8D12A12B12C12D12E12F24A24B24C24D
order1222222344444444666888812121212121224242424
size111112122422244668122224412124488884444

33 irreducible representations

dim1111111122222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C4○D4C4○D8C4○D12C8⋊C22S3×D4S3×D4Q83D6D24⋊C2
kernelD12.12D4C6.D8Dic3⋊C8C3×Q8⋊C4Dic35D4C2×D24C2×Q82S3C12.23D4Q8⋊C4D12C2×Dic3C4⋊C4C2×C8C2×Q8C12C6C4C6C4C22C2C2
# reps1111111112211124411122

Matrix representation of D12.12D4 in GL4(𝔽73) generated by

0100
72100
00722
00721
,
666600
59700
004132
005732
,
46000
04600
006112
006712
,
661400
59700
003241
00160
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,72,0,0,2,1],[66,59,0,0,66,7,0,0,0,0,41,57,0,0,32,32],[46,0,0,0,0,46,0,0,0,0,61,67,0,0,12,12],[66,59,0,0,14,7,0,0,0,0,32,16,0,0,41,0] >;

D12.12D4 in GAP, Magma, Sage, TeX

D_{12}._{12}D_4
% in TeX

G:=Group("D12.12D4");
// GroupNames label

G:=SmallGroup(192,378);
// by ID

G=gap.SmallGroup(192,378);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,184,1684,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^-1>;
// generators/relations

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