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G = C2×D4⋊D6order 192 = 26·3

Direct product of C2 and D4⋊D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D4⋊D6, C12.33C24, D12.29C23, C3⋊C85C23, C4○D417D6, (C2×D4)⋊41D6, (C2×Q8)⋊33D6, C65(C8⋊C22), D45(C22×S3), (C3×D4)⋊5C23, (C3×Q8)⋊5C23, Q86(C22×S3), D4⋊S318C22, (C2×C12).217D4, C12.426(C2×D4), (C6×D4)⋊45C22, C4.33(S3×C23), (C6×Q8)⋊37C22, (C2×D12)⋊58C22, (C22×D12)⋊20C2, (C22×C6).122D4, C6.158(C22×D4), (C22×C4).297D6, (C2×C12).555C23, Q82S317C22, C23.75(C3⋊D4), C4.Dic336C22, (C22×C12).290C22, C36(C2×C8⋊C22), (C6×C4○D4)⋊2C2, (C2×C4○D4)⋊6S3, (C2×D4⋊S3)⋊31C2, (C2×C3⋊C8)⋊22C22, (C2×C6).75(C2×D4), C4.29(C2×C3⋊D4), (C2×Q82S3)⋊31C2, (C3×C4○D4)⋊16C22, (C2×C4).95(C3⋊D4), (C2×C4.Dic3)⋊30C2, C2.31(C22×C3⋊D4), (C2×C4).245(C22×S3), C22.118(C2×C3⋊D4), SmallGroup(192,1379)

Series: Derived Chief Lower central Upper central

C1C12 — C2×D4⋊D6
C1C3C6C12D12C2×D12C22×D12 — C2×D4⋊D6
C3C6C12 — C2×D4⋊D6
C1C22C22×C4C2×C4○D4

Generators and relations for C2×D4⋊D6
 G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 872 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C3⋊C8, D12, D12, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×C6, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C2×C3⋊C8, C4.Dic3, D4⋊S3, Q82S3, C2×D12, C2×D12, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, S3×C23, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3, C2×Q82S3, D4⋊D6, C22×D12, C6×C4○D4, C2×D4⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C8⋊C22, C22×D4, C2×C3⋊D4, S3×C23, C2×C8⋊C22, D4⋊D6, C22×C3⋊D4, C2×D4⋊D6

Smallest permutation representation of C2×D4⋊D6
On 48 points
Generators in S48
(1 15)(2 13)(3 14)(4 21)(5 19)(6 20)(7 22)(8 23)(9 24)(10 16)(11 17)(12 18)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 11 5 8)(2 12 6 9)(3 10 4 7)(13 18 20 24)(14 16 21 22)(15 17 19 23)(25 33 28 36)(26 34 29 31)(27 35 30 32)(37 48 40 45)(38 43 41 46)(39 44 42 47)
(1 32)(2 36)(3 34)(4 31)(5 35)(6 33)(7 29)(8 27)(9 25)(10 26)(11 30)(12 28)(13 45)(14 43)(15 47)(16 38)(17 42)(18 40)(19 44)(20 48)(21 46)(22 41)(23 39)(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 14)(2 13)(3 15)(4 19)(5 21)(6 20)(7 17)(8 16)(9 18)(10 23)(11 22)(12 24)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F6A6B6C6D···6I8A8B8C8D12A12B12C12D12E···12J
order12222222222234444446666···688881212121212···12
size111122441212121222222442224···41212121222224···4

42 irreducible representations

dim111111122222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6D6D6C3⋊D4C3⋊D4C8⋊C22D4⋊D6
kernelC2×D4⋊D6C2×C4.Dic3C2×D4⋊S3C2×Q82S3D4⋊D6C22×D12C6×C4○D4C2×C4○D4C2×C12C22×C6C22×C4C2×D4C2×Q8C4○D4C2×C4C23C6C2
# reps112281113111146224

Matrix representation of C2×D4⋊D6 in GL6(𝔽73)

7200000
0720000
0072000
0007200
0000720
0000072
,
7200000
0720000
00661400
0059700
0000759
00001466
,
43600000
13300000
00006614
0000597
0075900
00146600
,
1720000
100000
0072100
0072000
0000172
000010
,
100000
1720000
0072000
0072100
00006614
000077

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,0,0,0,0,7,14,0,0,0,0,59,66],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,66,59,0,0,0,0,14,7,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;

C2×D4⋊D6 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_6
% in TeX

G:=Group("C2xD4:D6");
// GroupNames label

G:=SmallGroup(192,1379);
// by ID

G=gap.SmallGroup(192,1379);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,1684,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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