Copied to
clipboard

?

G = D12⋊20D4  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊20D4, C6.412+ (1+4), C4⋊C4⋊24D6, (C2×D4)⋊8D6, D6⋊9(C4○D4), C4⋊D4⋊15S3, C3⋊6(D4⋊5D4), C22⋊C4⋊28D6, D6.19(C2×D4), C4.110(S3×D4), (C22×C4)⋊22D6, C23⋊2D6⋊11C2, D6⋊3D4⋊21C2, D6⋊C4⋊18C22, C4.D12⋊22C2, C12.229(C2×D4), (C6×D4)⋊14C22, Dic3⋊5D4⋊21C2, C6.71(C22×D4), C23.9D6⋊20C2, (C2×C6).156C24, C4⋊Dic3⋊32C22, C2.43(D4⋊6D6), C23.12D6⋊17C2, (C2×C12).595C23, Dic3⋊C4⋊65C22, (C22×C12)⋊22C22, (C2×Dic6)⋊62C22, (C4×Dic3)⋊23C22, (C22×C6).23C23, (C2×D12).264C22, C6.D4⋊24C22, (S3×C23).48C22, C22.177(S3×C23), C23.123(C22×S3), (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

Derived series
C1 — C2×C6 — D12⋊20D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊20D4
Lower central
C3 — C2×C6 — D12⋊20D4

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], D4⋊5D4, S3×C22⋊C4 [×2], C23.9D6 [×2], Dic3⋊5D4, C4.D12, C4×C3⋊D4, C23.12D6, C23⋊2D6 [×2], D6⋊3D4 [×2], C3×C4⋊D4, C2×C4○D12, C2×S3×D4, D12⋊20D4

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, D4⋊5D4, C2×S3×D4, D4⋊6D6, S3×C4○D4, D12⋊20D4

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

Smallest permutation representation
►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 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)

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

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

Matrix representation ►G ⊆ 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] >;
 

39 conjugacy classes

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

39 irreducible representations

dim11111111111122222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4○D42+ (1+4)S3×D4D4⋊6D6S3×C4○D4
kernelD12⋊20D4S3×C22⋊C4C23.9D6Dic3⋊5D4C4.D12C4×C3⋊D4C23.12D6C23⋊2D6D6⋊3D4C3×C4⋊D4C2×C4○D12C2×S3×D4C4⋊D4D12C22⋊C4C4⋊C4C22×C4C2×D4D6C6C4C2C2
# reps12211112211114211341222

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
 

׿
×
⋊
ℤ
𝔽
○
≀