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G = D1220D4order 192 = 26·3

8th semidirect product of D12 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1220D4, C6.412+ 1+4, C4⋊C424D6, (C2×D4)⋊8D6, D69(C4○D4), C4⋊D415S3, C36(D45D4), C22⋊C428D6, D6.19(C2×D4), C4.110(S3×D4), (C22×C4)⋊22D6, D63D421C2, C232D611C2, D6⋊C418C22, C4.D1222C2, C12.229(C2×D4), (C6×D4)⋊14C22, Dic35D421C2, C6.71(C22×D4), C23.9D620C2, (C2×C6).156C24, C4⋊Dic332C22, C2.43(D46D6), C23.12D617C2, (C2×C12).595C23, Dic3⋊C465C22, (C22×C12)⋊22C22, (C4×Dic3)⋊23C22, (C2×Dic6)⋊62C22, (C22×C6).23C23, (C2×D12).264C22, C6.D424C22, (S3×C23).48C22, C23.123(C22×S3), C22.177(S3×C23), (C2×Dic3).75C23, (C22×S3).190C23, (C2×S3×D4)⋊13C2, C2.44(C2×S3×D4), (C4×C3⋊D4)⋊18C2, (S3×C22⋊C4)⋊6C2, (S3×C2×C4)⋊15C22, C2.40(S3×C4○D4), (C2×C4○D12)⋊22C2, (C3×C4⋊D4)⋊18C2, (C3×C4⋊C4)⋊13C22, C6.153(C2×C4○D4), (C2×C3⋊D4)⋊40C22, (C2×C4).39(C22×S3), (C3×C22⋊C4)⋊15C22, SmallGroup(192,1171)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D1220D4
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — D1220D4
C3C2×C6 — D1220D4
C1C22C4⋊D4

Generators and relations for D1220D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 992 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×29], S3 [×6], C6 [×3], C6 [×3], C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×18], Q8 [×2], C23, C23 [×2], C23 [×13], Dic3 [×5], C12 [×2], C12 [×3], D6 [×6], D6 [×14], C2×C6, C2×C6 [×9], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×8], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×10], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3, C22×S3 [×2], C22×S3 [×10], C22×C6, C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×4], C6.D4, C6.D4 [×4], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4 [×4], C2×D12, C4○D12 [×4], S3×D4 [×4], C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4, C6×D4 [×2], S3×C23 [×2], D45D4, S3×C22⋊C4 [×2], C23.9D6 [×2], Dic35D4, C4.D12, C4×C3⋊D4, C23.12D6, C232D6 [×2], D63D4 [×2], C3×C4⋊D4, C2×C4○D12, C2×S3×D4, D1220D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, S3×D4 [×2], S3×C23, D45D4, C2×S3×D4, D46D6, S3×C4○D4, D1220D4

Smallest permutation representation of D1220D4
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 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)
(1 21 39 32)(2 16 40 27)(3 23 41 34)(4 18 42 29)(5 13 43 36)(6 20 44 31)(7 15 45 26)(8 22 46 33)(9 17 47 28)(10 24 48 35)(11 19 37 30)(12 14 38 25)
(1 39)(2 46)(3 41)(4 48)(5 43)(6 38)(7 45)(8 40)(9 47)(10 42)(11 37)(12 44)(14 20)(16 22)(18 24)(25 31)(27 33)(29 35)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G···2L 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E6F6G12A12B12C12D12E12F
order12222222···234444444444446666666121212121212
size11114446···6222224466121212122224488444488

39 irreducible representations

dim11111111111122222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4○D42+ 1+4S3×D4D46D6S3×C4○D4
kernelD1220D4S3×C22⋊C4C23.9D6Dic35D4C4.D12C4×C3⋊D4C23.12D6C232D6D63D4C3×C4⋊D4C2×C4○D12C2×S3×D4C4⋊D4D12C22⋊C4C4⋊C4C22×C4C2×D4D6C6C4C2C2
# reps12211112211114211341222

Matrix representation of D1220D4 in GL6(𝔽13)

100000
010000
001100
0012000
000008
000080
,
100000
010000
001100
0001200
000008
000050
,
010000
1200000
0012000
0001200
000001
0000120
,
1200000
010000
001000
000100
0000120
000001

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,5,0,0,0,0,8,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;

D1220D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{20}D_4
% in TeX

G:=Group("D12:20D4");
// GroupNames label

G:=SmallGroup(192,1171);
// by ID

G=gap.SmallGroup(192,1171);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,6278]);
// 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=d*a*d=a^7,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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