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G = D1219D4order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1219D4, C6.1142+ 1+4, C33D42, C43(S3×D4), C4⋊C423D6, D67(C2×D4), C126(C2×D4), C3⋊D41D4, (C2×D4)⋊23D6, C223(S3×D4), C4⋊D412S3, C22⋊C411D6, Dic34(C2×D4), (C22×C4)⋊20D6, D6⋊D413C2, C12⋊D422C2, Dic3⋊D420C2, C232D610C2, C123D417C2, D6⋊C453C22, (C6×D4)⋊13C22, Dic35D420C2, C6.68(C22×D4), C2.28(D4○D12), (C22×D12)⋊15C2, (C2×D12)⋊46C22, (C2×C6).153C24, (C2×C12).40C23, Dic3⋊C452C22, (S3×C23)⋊10C22, (C22×C12)⋊21C22, (C4×Dic3)⋊22C22, C6.D451C22, (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

C1C2×C6 — D1219D4
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — D1219D4
C3C2×C6 — D1219D4
C1C22C4⋊D4

Generators and relations for D1219D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=dbd=a6b, dcd=c-1 >

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], C41D4, 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], Dic35D4, C12⋊D4, C4×C3⋊D4, C232D6 [×2], C123D4, C3×C4⋊D4, C22×D12, C2×S3×D4, C2×S3×D4 [×2], D1219D4
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, D1219D4

Smallest permutation representation of D1219D4
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 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)
(1 38 35 16)(2 45 36 23)(3 40 25 18)(4 47 26 13)(5 42 27 20)(6 37 28 15)(7 44 29 22)(8 39 30 17)(9 46 31 24)(10 41 32 19)(11 48 33 14)(12 43 34 21)
(1 29)(2 36)(3 31)(4 26)(5 33)(6 28)(7 35)(8 30)(9 25)(10 32)(11 27)(12 34)(14 20)(16 22)(18 24)(38 44)(40 46)(42 48)

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim1111111111122222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2S3D4D4D6D6D6D62+ 1+4S3×D4S3×D4D4○D12
kernelD1219D4D6⋊D4Dic3⋊D4Dic35D4C12⋊D4C4×C3⋊D4C232D6C123D4C3×C4⋊D4C22×D12C2×S3×D4C4⋊D4D12C3⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C6C4C22C2
# reps1221112111314421131222

Matrix representation of D1219D4 in 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] >;

D1219D4 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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