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G = D45D12order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D45D12, C4217D6, C6.172+ 1+4, C4⋊C448D6, (C3×D4)⋊10D4, (C4×D4)⋊16S3, D67(C4○D4), (C4×D12)⋊30C2, D6⋊D46C2, (D4×C12)⋊18C2, C127D49C2, C33(D45D4), C22⋊C447D6, C12.54(C2×D4), C4.22(C2×D12), (C22×C4)⋊16D6, C12⋊D415C2, (C4×C12)⋊20C22, D6⋊C452C22, C4.D1214C2, (C2×D4).248D6, (C2×D12)⋊6C22, (C2×C6).98C24, C4⋊Dic38C22, C22.1(C2×D12), C6.16(C22×D4), C427S318C2, C2.18(C22×D12), C2.18(D46D6), (C2×C12).159C23, (C22×C12)⋊10C22, (C2×Dic6)⋊17C22, (C6×D4).259C22, C23.21D65C2, (C22×S3).33C23, (S3×C23).40C22, C22.123(S3×C23), C23.182(C22×S3), (C22×C6).168C23, (C2×Dic3).42C23, (C22×Dic3)⋊9C22, (C2×S3×D4)⋊4C2, (S3×C2×C4)⋊3C22, (C2×C6).1(C2×D4), (C2×D6⋊C4)⋊21C2, C2.22(S3×C4○D4), (C2×D42S3)⋊3C2, (C3×C4⋊C4)⋊60C22, C6.139(C2×C4○D4), (C2×C3⋊D4)⋊4C22, (C3×C22⋊C4)⋊50C22, (C2×C4).160(C22×S3), SmallGroup(192,1113)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D45D12
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — D45D12
C3C2×C6 — D45D12
C1C22C4×D4

Generators and relations for D45D12
 G = < a,b,c,d | a4=b2=c12=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1016 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], S3 [×5], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×14], Dic3 [×4], C12 [×2], C12 [×4], D6 [×2], D6 [×19], C2×C6, C2×C6 [×4], C2×C6 [×4], 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 [×4], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3 [×2], C22×S3 [×2], 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, C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×8], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×2], S3×D4 [×4], D42S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23 [×2], D45D4, C4×D12, C427S3, D6⋊D4 [×2], C23.21D6 [×2], C12⋊D4, C4.D12, C2×D6⋊C4 [×2], C127D4 [×2], D4×C12, C2×S3×D4, C2×D42S3, D45D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C2×D12 [×6], S3×C23, D45D4, C22×D12, D46D6, S3×C4○D4, D45D12

Smallest permutation representation of D45D12
On 48 points
Generators in S48
(1 37 24 34)(2 35 13 38)(3 39 14 36)(4 25 15 40)(5 41 16 26)(6 27 17 42)(7 43 18 28)(8 29 19 44)(9 45 20 30)(10 31 21 46)(11 47 22 32)(12 33 23 48)
(1 34)(2 38)(3 36)(4 40)(5 26)(6 42)(7 28)(8 44)(9 30)(10 46)(11 32)(12 48)(13 35)(14 39)(15 25)(16 41)(17 27)(18 43)(19 29)(20 45)(21 31)(22 47)(23 33)(24 37)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E6F6G12A12B12C12D12E···12L
order1222222222222344444444444466666661212121212···12
size11112222661212122222244466121212222444422224···4

45 irreducible representations

dim111111111111222222222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6D6C4○D4D122+ 1+4D46D6S3×C4○D4
kernelD45D12C4×D12C427S3D6⋊D4C23.21D6C12⋊D4C4.D12C2×D6⋊C4C127D4D4×C12C2×S3×D4C2×D42S3C4×D4C3×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D6D4C6C2C2
# reps111221122111141212148122

Matrix representation of D45D12 in GL6(𝔽13)

100000
010000
001000
000100
000053
000008
,
1200000
0120000
001000
000100
000053
000058
,
110000
1200000
000100
0012000
000080
000085
,
12120000
010000
000100
001000
0000122
000001

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,3,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,3,8],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,2,1] >;

D45D12 in GAP, Magma, Sage, TeX

D_4\rtimes_5D_{12}
% in TeX

G:=Group("D4:5D12");
// GroupNames label

G:=SmallGroup(192,1113);
// by ID

G=gap.SmallGroup(192,1113);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,192,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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