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G = D1223D4order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1223D4, C4216D6, C6.1032+ 1+4, C4⋊C447D6, (C4×D4)⋊12S3, (C4×D12)⋊28C2, (D4×C12)⋊14C2, Dic3⋊D47C2, C32(D45D4), C22⋊C446D6, D6.14(C2×D4), C4.139(S3×D4), (C22×C4)⋊15D6, C232D620C2, C127D418C2, (C4×C12)⋊19C22, D6⋊C430C22, D6.D47C2, (C2×D4).213D6, C12.345(C2×D4), (C22×D12)⋊9C2, (C2×C6).94C24, C6.49(C22×D4), C427S316C2, C2.15(D4○D12), C224(C4○D12), Dic3⋊C43C22, C4⋊Dic359C22, C12.48D410C2, (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

C1C2×C6 — D1223D4
C1C3C6C2×C6C22×S3S3×C23S3×C22⋊C4 — D1223D4
C3C2×C6 — D1223D4
C1C22C4×D4

Generators and relations for D1223D4
 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 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], S3 [×6], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×4], C12 [×2], C12 [×4], D6 [×4], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×6], D12 [×4], D12 [×6], C2×Dic3 [×4], C3⋊D4 [×6], C2×C12 [×5], C2×C12 [×4], C3×D4 [×2], C22×S3 [×4], C22×S3 [×10], C22×C6 [×2], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×8], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×2], C2×D12 [×4], C4○D12 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23 [×2], D45D4, C4×D12, C427S3, S3×C22⋊C4 [×2], Dic3⋊D4 [×2], D6.D4 [×2], C12.48D4, C127D4, C232D6 [×2], D4×C12, C22×D12, C2×C4○D12, D1223D4
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, C4○D12 [×2], S3×D4 [×2], S3×C23, D45D4, C2×C4○D12, C2×S3×D4, D4○D12, D1223D4

Smallest permutation representation of D1223D4
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 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)
(1 19 41 36)(2 20 42 25)(3 21 43 26)(4 22 44 27)(5 23 45 28)(6 24 46 29)(7 13 47 30)(8 14 48 31)(9 15 37 32)(10 16 38 33)(11 17 39 34)(12 18 40 35)
(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)

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

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

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

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
kernelD1223D4C4×D12C427S3S3×C22⋊C4Dic3⋊D4D6.D4C12.48D4C127D4C232D6D4×C12C22×D12C2×C4○D12C4×D4D12C42C22⋊C4C4⋊C4C22×C4C2×D4C2×C6C22C6C4C2
# reps111222112111141212148122

Matrix representation of D1223D4 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] >;

D1223D4 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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