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G = D12⋊23D4  order 192 = 26·3

1st semidirect product of D12 and D4 acting through Inn(D12)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12⋊23D4, C42⋊16D6, C6.1032+ 1+4, C4⋊C4⋊47D6, (C4×D4)⋊12S3, (C4×D12)⋊28C2, (D4×C12)⋊14C2, Dic3⋊D4⋊7C2, C3⋊2(D4⋊5D4), C22⋊C4⋊46D6, D6.14(C2×D4), C4.139(S3×D4), (C22×C4)⋊15D6, C23⋊2D6⋊20C2, C12⋊7D4⋊18C2, (C4×C12)⋊19C22, D6⋊C4⋊30C22, D6.D4⋊7C2, (C2×D4).213D6, C12.345(C2×D4), (C22×D12)⋊9C2, (C2×C6).94C24, C6.49(C22×D4), C42⋊7S3⋊16C2, C2.15(D4○D12), C22⋊4(C4○D12), Dic3⋊C4⋊3C22, C4⋊Dic3⋊59C22, C12.48D4⋊10C2, (C2×C12).782C23, (C22×C12)⋊16C22, (C2×Dic6)⋊53C22, (C6×D4).305C22, (C2×D12).210C22, (S3×C23).39C22, C23.104(C22×S3), C22.119(S3×C23), (C22×C6).164C23, (C2×Dic3).40C23, (C22×S3).172C23, C6.D4.11C22, C2.22(C2×S3×D4), (C2×C4○D12)⋊7C2, (C2×C6)⋊2(C4○D4), (S3×C2×C4)⋊48C22, C6.41(C2×C4○D4), (C3×C4⋊C4)⋊59C22, (S3×C22⋊C4)⋊28C2, C2.45(C2×C4○D12), (C2×C3⋊D4)⋊3C22, (C3×C22⋊C4)⋊56C22, (C2×C4).158(C22×S3), SmallGroup(192,1109)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D12⋊23D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — D12⋊23D4
Lower central
C3 — C2×C6 — D12⋊23D4
Upper central
C1 — C22 — C4×D4

Generators and relations for D12⋊23D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 1016 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, D4⋊5D4, C4×D12, C42⋊7S3, S3×C22⋊C4, Dic3⋊D4, D6.D4, C12.48D4, C12⋊7D4, C23⋊2D6, D4×C12, C22×D12, C2×C4○D12, D12⋊23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, C4○D12, S3×D4, S3×C23, D4⋊5D4, C2×C4○D12, C2×S3×D4, D4○D12, D12⋊23D4

Smallest permutation representation of D12⋊23D4
►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 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A···4F4G4H4I4J4K4L6A6B6C6D6E6F6G12A12B12C12D12E···12L
order122222222222234···444444466666661212121212···12
size11112246666121222···24412121212222444422224···4

45 irreducible representations

dim111111111111222222222444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6D6C4○D4C4○D122+ 1+4S3×D4D4○D12
kernelD12⋊23D4C4×D12C42⋊7S3S3×C22⋊C4Dic3⋊D4D6.D4C12.48D4C12⋊7D4C23⋊2D6D4×C12C22×D12C2×C4○D12C4×D4D12C42C22⋊C4C4⋊C4C22×C4C2×D4C2×C6C22C6C4C2
# reps111222112111141212148122

Matrix representation of D12⋊23D4 ►in GL4(𝔽13) generated by

12000
01200
00310
0036
,
12000
01200
00310
00710
,
01200
1000
0024
00911
,
1000
01200
0010
0001
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,3,3,0,0,10,6],[12,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10],[0,1,0,0,12,0,0,0,0,0,2,9,0,0,4,11],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
 

D12⋊23D4 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_{23}D_4
 
% in TeX
 
G:=Group("D12:23D4");
 
// GroupNames label
 
G:=SmallGroup(192,1109);
 
// by ID
 
G=gap.SmallGroup(192,1109);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,570,6278]);
 
// 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,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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