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G = D12⋊11D4  order 192 = 26·3

4th semidirect product of D12 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊11D4, C42⋊29D6, C6.772+ (1+4), C3⋊4(D42), C4⋊2(S3×D4), D6⋊8(C2×D4), C12⋊3(C2×D4), (C2×D4)⋊26D6, C4⋊1D4⋊7S3, (C4×D12)⋊49C2, C23⋊2D6⋊27C2, D6⋊3D4⋊36C2, (C4×C12)⋊27C22, D6⋊C4⋊70C22, (C6×D4)⋊33C22, C6.95(C22×D4), (C2×C6).261C24, C4⋊Dic3⋊74C22, C2.81(D4⋊6D6), (C2×C12).509C23, (S3×C23)⋊13C22, (C22×C6).75C23, C23.77(C22×S3), (C2×D12).269C22, C6.D4⋊37C22, C22.282(S3×C23), (C22×S3).229C23, (C2×Dic3).136C23, (C2×S3×D4)⋊20C2, C2.68(C2×S3×D4), (C3×C4⋊1D4)⋊8C2, (S3×C2×C4)⋊29C22, (C2×C3⋊D4)⋊27C22, (C2×C4).214(C22×S3), SmallGroup(192,1276)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D12⋊11D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D12⋊11D4
Lower central
C3 — C2×C6 — D12⋊11D4

Subgroups: 1376 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×4], C4 [×5], C22, C22 [×44], S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×4], C23 [×24], Dic3 [×4], C12 [×4], C12, D6 [×8], D6 [×24], C2×C6, C2×C6 [×12], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×D4 [×26], C24 [×4], C4×S3 [×8], D12 [×8], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12, C2×C12 [×2], C3×D4 [×10], C22×S3 [×4], C22×S3 [×20], C22×C6 [×4], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×4], C4×C12, S3×C2×C4 [×4], C2×D12 [×2], S3×D4 [×16], C2×C3⋊D4 [×8], C6×D4 [×6], S3×C23 [×4], D42, C4×D12 [×2], C23⋊2D6 [×4], D6⋊3D4 [×4], C3×C4⋊1D4, C2×S3×D4 [×4], D12⋊11D4

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⋊6D6, D12⋊11D4

Generators and relations
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, dad=a7, cbc-1=a6b, bd=db, 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 43)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 48)(9 47)(10 46)(11 45)(12 44)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
000100
0012100
0000710
000086
,
1200000
0120000
0012100
000100
000063
0000107
,
1230000
810000
001000
000100
000063
000057
,
1200000
810000
0012000
0001200
0000710
000036

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,5,0,0,0,0,3,7],[12,8,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2O 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G12A···12F
order122222222···23444444444666666612···12
size111144446···62222241212121222288884···4

39 irreducible representations

dim1111112222444
type++++++++++++
imageC1C2C2C2C2C2S3D4D6D62+ (1+4)S3×D4D4⋊6D6
kernelD12⋊11D4C4×D12C23⋊2D6D6⋊3D4C3×C4⋊1D4C2×S3×D4C4⋊1D4D12C42C2×D4C6C4C2
# reps1244141816142

In GAP, Magma, Sage, TeX 

D_{12}\rtimes_{11}D_4
 
% in TeX
 
G:=Group("D12:11D4");
 
// GroupNames label
 
G:=SmallGroup(192,1276);
 
// by ID
 
G=gap.SmallGroup(192,1276);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,136,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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