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G = D12.36D4order 192 = 26·3

6th non-split extension by D12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.36D4, C4:C4:4D6, (C2xQ8):4D6, (C2xC6):5SD16, C22:Q8:1S3, (C2xC12).77D4, C4.101(S3xD4), (C6xQ8):1C22, C6.47C22wrC2, C12.152(C2xD4), C6.D8:37C2, C3:3(C22:SD16), C6.72(C2xSD16), (C22xC6).92D4, (C22xC4).143D6, C2.15(C23:2D6), C2.15(D4:D6), C12.55D4:13C2, C6.116(C8:C22), (C2xC12).365C23, C22:3(Q8:2S3), (C22xD12).13C2, C23.69(C3:D4), (C2xD12).242C22, (C22xC12).169C22, (C2xC3:C8):6C22, (C3xC4:C4):6C22, (C3xC22:Q8):1C2, (C2xQ8:2S3):8C2, (C2xC6).496(C2xD4), C2.9(C2xQ8:2S3), (C2xC4).55(C3:D4), (C2xC4).465(C22xS3), C22.171(C2xC3:D4), SmallGroup(192,605)

Series: Derived Chief Lower central Upper central

C1C2xC12 — D12.36D4
C1C3C6C12C2xC12C2xD12C22xD12 — D12.36D4
C3C6C2xC12 — D12.36D4
C1C22C22xC4C22:Q8

Generators and relations for D12.36D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd=a6c-1 >

Subgroups: 704 in 188 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C12, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C24, C3:C8, D12, D12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xC6, C22:C8, D4:C4, C22:Q8, C2xSD16, C22xD4, C2xC3:C8, Q8:2S3, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xD12, C2xD12, C22xC12, C6xQ8, S3xC23, C22:SD16, C6.D8, C12.55D4, C2xQ8:2S3, C3xC22:Q8, C22xD12, D12.36D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C3:D4, C22xS3, C22wrC2, C2xSD16, C8:C22, Q8:2S3, S3xD4, C2xC3:D4, C22:SD16, C23:2D6, C2xQ8:2S3, D4:D6, D12.36D4

Smallest permutation representation of D12.36D4
On 48 points
Generators in S48
(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)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 15)(16 24)(17 23)(18 22)(19 21)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 47)(38 46)(39 45)(40 44)(41 43)
(1 44 31 16)(2 39 32 23)(3 46 33 18)(4 41 34 13)(5 48 35 20)(6 43 36 15)(7 38 25 22)(8 45 26 17)(9 40 27 24)(10 47 28 19)(11 42 29 14)(12 37 30 21)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)

G:=sub<Sym(48)| (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,15)(16,24)(17,23)(18,22)(19,21)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,47)(38,46)(39,45)(40,44)(41,43), (1,44,31,16)(2,39,32,23)(3,46,33,18)(4,41,34,13)(5,48,35,20)(6,43,36,15)(7,38,25,22)(8,45,26,17)(9,40,27,24)(10,47,28,19)(11,42,29,14)(12,37,30,21), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)>;

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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,15)(16,24)(17,23)(18,22)(19,21)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,47)(38,46)(39,45)(40,44)(41,43), (1,44,31,16)(2,39,32,23)(3,46,33,18)(4,41,34,13)(5,48,35,20)(6,43,36,15)(7,38,25,22)(8,45,26,17)(9,40,27,24)(10,47,28,19)(11,42,29,14)(12,37,30,21), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36) );

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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,15),(16,24),(17,23),(18,22),(19,21),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,47),(38,46),(39,45),(40,44),(41,43)], [(1,44,31,16),(2,39,32,23),(3,46,33,18),(4,41,34,13),(5,48,35,20),(6,43,36,15),(7,38,25,22),(8,45,26,17),(9,40,27,24),(10,47,28,19),(11,42,29,14),(12,37,30,21)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)]])

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
order12222222223444446666688881212121212121212
size11112212121212222488222441212121244448888

33 irreducible representations

dim11111122222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6D6SD16C3:D4C3:D4C8:C22S3xD4Q8:2S3D4:D6
kernelD12.36D4C6.D8C12.55D4C2xQ8:2S3C3xC22:Q8C22xD12C22:Q8D12C2xC12C22xC6C4:C4C22xC4C2xQ8C2xC6C2xC4C23C6C4C22C2
# reps12121114111114221222

Matrix representation of D12.36D4 in GL6(F73)

110000
7200000
0072000
0007200
000001
0000720
,
110000
0720000
0072000
0072100
000001
000010
,
43130000
60300000
0072200
0072100
00006767
0000676
,
100000
010000
001000
0017200
0000720
0000072

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,67,67,0,0,0,0,67,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

D12.36D4 in GAP, Magma, Sage, TeX

D_{12}._{36}D_4
% in TeX

G:=Group("D12.36D4");
// GroupNames label

G:=SmallGroup(192,605);
// by ID

G=gap.SmallGroup(192,605);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,1123,297,136,6278]);
// Polycyclic

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

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