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## G = C2×Q8⋊3D6order 192 = 26·3

### Direct product of C2 and Q8⋊3D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q8⋊3D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — C2×Q8⋊3D6
 Lower central C3 — C6 — C12 — C2×Q8⋊3D6
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×Q83D6
G = < a,b,c,d,e | a2=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 952 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C22×C6, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C8⋊S3, D24, C2×C3⋊C8, D4⋊S3, Q82S3, C2×C24, C3×SD16, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, S3×D4, Q83S3, Q83S3, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C2×C8⋊C22, C2×C8⋊S3, C2×D24, Q83D6, C2×D4⋊S3, C2×Q82S3, C6×SD16, C2×S3×D4, C2×Q83S3, C2×Q83D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C8⋊C22, C22×D4, S3×D4, S3×C23, C2×C8⋊C22, Q83D6, C2×S3×D4, C2×Q83D6

Smallest permutation representation of C2×Q83D6
On 48 points
Generators in S48
(1 11)(2 12)(3 10)(4 9)(5 7)(6 8)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 23 4 20)(2 21 5 24)(3 19 6 22)(7 15 12 18)(8 13 10 16)(9 17 11 14)(25 45 48 28)(26 29 43 46)(27 47 44 30)(31 37 40 34)(32 35 41 38)(33 39 42 36)
(1 30 4 47)(2 28 5 45)(3 26 6 43)(7 36 12 39)(8 34 10 37)(9 32 11 41)(13 40 16 31)(14 35 17 38)(15 42 18 33)(19 46 22 29)(20 27 23 44)(21 48 24 25)
(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 3)(4 6)(8 9)(10 11)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 4 4 6 6 12 12 12 12 2 2 2 4 4 6 6 2 2 2 8 8 4 4 12 12 4 4 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 C8⋊C22 S3×D4 S3×D4 Q8⋊3D6 kernel C2×Q8⋊3D6 C2×C8⋊S3 C2×D24 Q8⋊3D6 C2×D4⋊S3 C2×Q8⋊2S3 C6×SD16 C2×S3×D4 C2×Q8⋊3S3 C2×SD16 C4×S3 C2×Dic3 C22×S3 C2×C8 SD16 C2×D4 C2×Q8 C6 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 1 2 1 1 1 4 1 1 2 1 1 4

Matrix representation of C2×Q83D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 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
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 1 0 0 72 72 71 72 0 0 25 25 1 0 0 0 24 25 1 0
,
 72 70 0 0 0 0 0 1 0 0 0 0 0 0 39 10 44 44 0 0 34 5 0 44 0 0 68 68 68 63 0 0 39 68 39 34
,
 72 0 0 0 0 0 25 1 0 0 0 0 0 0 1 1 1 2 0 0 0 0 72 1 0 0 72 0 0 72 0 0 0 72 0 72
,
 1 0 0 0 0 0 48 72 0 0 0 0 0 0 0 0 1 72 0 0 72 72 72 71 0 0 1 0 0 1 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,25,24,0,0,0,72,25,25,0,0,72,71,1,1,0,0,1,72,0,0],[72,0,0,0,0,0,70,1,0,0,0,0,0,0,39,34,68,39,0,0,10,5,68,68,0,0,44,0,68,39,0,0,44,44,63,34],[72,25,0,0,0,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,1,0,0,72,0,0,1,72,0,0,0,0,2,1,72,72],[1,48,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,72,71,1,1] >;

C2×Q83D6 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3D_6
% in TeX

G:=Group("C2xQ8:3D6");
// GroupNames label

G:=SmallGroup(192,1318);
// by ID

G=gap.SmallGroup(192,1318);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,185,136,438,235,102,6278]);
// Polycyclic

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

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