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G = D12⋊19D4  order 192 = 26·3

7th semidirect product of D12 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊19D4, C6.1142+ (1+4), C3⋊3(D42), C4⋊3(S3×D4), C4⋊C4⋊23D6, D6⋊7(C2×D4), C12⋊6(C2×D4), C3⋊D4⋊1D4, (C2×D4)⋊23D6, C22⋊3(S3×D4), C4⋊D4⋊12S3, C22⋊C4⋊11D6, Dic3⋊4(C2×D4), (C22×C4)⋊20D6, D6⋊D4⋊13C2, C12⋊D4⋊22C2, Dic3⋊D4⋊20C2, C23⋊2D6⋊10C2, C12⋊3D4⋊17C2, D6⋊C4⋊53C22, (C6×D4)⋊13C22, Dic3⋊5D4⋊20C2, C6.68(C22×D4), C2.28(D4○D12), (C2×D12)⋊46C22, (C22×D12)⋊15C2, (C2×C12).40C23, (C2×C6).153C24, Dic3⋊C4⋊52C22, (S3×C23)⋊10C22, (C22×C12)⋊21C22, (C4×Dic3)⋊22C22, C6.D4⋊51C22, (C22×S3).64C23, C23.191(C22×S3), (C22×C6).188C23, C22.174(S3×C23), (C2×Dic3).227C23, (C2×S3×D4)⋊11C2, (C2×C6)⋊3(C2×D4), C2.41(C2×S3×D4), (C4×C3⋊D4)⋊16C2, (S3×C2×C4)⋊14C22, (C3×C4⋊D4)⋊15C2, (C3×C4⋊C4)⋊11C22, (C2×C3⋊D4)⋊15C22, (C3×C22⋊C4)⋊13C22, (C2×C4).176(C22×S3), SmallGroup(192,1168)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D12⋊19D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊19D4
Lower central
C3 — C2×C6 — D12⋊19D4

Subgroups: 1408 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×2], C4 [×7], C22, C22 [×2], C22 [×42], S3 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×11], D4 [×34], C23, C23 [×2], C23 [×25], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×3], D6 [×6], D6 [×28], C2×C6, C2×C6 [×2], C2×C6 [×8], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×29], C24 [×4], C4×S3 [×6], D12 [×4], D12 [×10], C2×Dic3 [×3], C3⋊D4 [×4], C3⋊D4 [×10], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×S3, C22×S3 [×4], C22×S3 [×20], C22×C6, C22×C6 [×2], C4×D4 [×2], C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C4⋊1D4, C22×D4 [×4], C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4 [×4], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×4], C2×D12 [×4], S3×D4 [×12], C2×C3⋊D4, C2×C3⋊D4 [×6], C22×C12, C6×D4, C6×D4 [×2], S3×C23 [×4], D42, D6⋊D4 [×2], Dic3⋊D4 [×2], Dic3⋊5D4, C12⋊D4, C4×C3⋊D4, C23⋊2D6 [×2], C12⋊3D4, C3×C4⋊D4, C22×D12, C2×S3×D4, C2×S3×D4 [×2], D12⋊19D4

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C24, C22×S3 [×7], C22×D4 [×2], 2+ (1+4), S3×D4 [×4], S3×C23, D42, C2×S3×D4 [×2], D4○D12, D12⋊19D4

Generators and relations
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=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 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)

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

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

Matrix representation ►G ⊆ GL6(ℤ)

010000
-100000
000100
00-1100
000010
000001
,
010000
100000
00-1100
000100
000010
000001
,
-100000
010000
001000
000100
000001
0000-10
,
-100000
010000
001000
000100
0000-10
000001

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2M2N2O 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G12A12B12C12D12E12F
order122222222···22234444444446666666121212121212
size111122446···612122224446612122224488444488

39 irreducible representations

dim1111111111122222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D4D6D6D6D62+ (1+4)S3×D4S3×D4D4○D12
kernelD12⋊19D4D6⋊D4Dic3⋊D4Dic3⋊5D4C12⋊D4C4×C3⋊D4C23⋊2D6C12⋊3D4C3×C4⋊D4C22×D12C2×S3×D4C4⋊D4D12C3⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C6C4C22C2
# reps1221112111314421131222

In GAP, Magma, Sage, TeX 

D_{12}\rtimes_{19}D_4
 
% in TeX
 
G:=Group("D12:19D4");
 
// GroupNames label
 
G:=SmallGroup(192,1168);
 
// by ID
 
G=gap.SmallGroup(192,1168);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,192,6278]);
 
// 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,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;
 
// generators/relations
 

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