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## G = C2×U2(𝔽3)  order 192 = 26·3

### Direct product of C2 and U2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×U2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — U2(𝔽3) — C2×U2(𝔽3)
 Lower central SL2(𝔽3) — C2×U2(𝔽3)
 Upper central C1 — C2×C4

Generators and relations for C2×U2(𝔽3)
G = < a,b,c,d,e,f | a2=b4=e3=1, c2=d2=b2, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >

Subgroups: 275 in 81 conjugacy classes, 21 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), C2×C12, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C2×SL2(𝔽3), C4.A4, C2×C4≀C2, U2(𝔽3), C2×C4.A4, C2×U2(𝔽3)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, U2(𝔽3), C2×A4⋊C4, C2×U2(𝔽3)

Smallest permutation representation of C2×U2(𝔽3)
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)
(1 16 5 12)(2 15 6 11)(3 10 7 14)(4 9 8 13)(17 19 21 23)(18 31 22 27)(20 25 24 29)(26 32 30 28)(33 39 37 35)(34 43 38 47)(36 45 40 41)(42 44 46 48)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 28 21 32)(18 25 22 29)(19 30 23 26)(20 27 24 31)(33 44 37 48)(34 41 38 45)(35 46 39 42)(36 43 40 47)
(1 38 17)(2 18 39)(3 40 19)(4 20 33)(5 34 21)(6 22 35)(7 36 23)(8 24 37)(9 27 46)(10 47 28)(11 29 48)(12 41 30)(13 31 42)(14 43 32)(15 25 44)(16 45 26)
(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)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E ··· 4N 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D order 1 2 2 2 2 2 3 4 4 4 4 4 ··· 4 6 6 6 8 8 8 8 12 12 12 12 size 1 1 1 1 6 6 8 1 1 1 1 6 ··· 6 8 8 8 12 12 12 12 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 type + + + + - - + + + image C1 C2 C2 C4 C4 S3 Dic3 Dic3 D6 U2(𝔽3) S4 A4⋊C4 C2×S4 A4⋊C4 U2(𝔽3) kernel C2×U2(𝔽3) U2(𝔽3) C2×C4.A4 C2×SL2(𝔽3) C4.A4 C2×C4○D4 C2×Q8 C4○D4 C4○D4 C2 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 1 1 1 1 8 2 2 2 2 4

Matrix representation of C2×U2(𝔽3) in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 46 0 0 0 0 46
,
 1 0 0 0 0 1 0 0 0 0 1 71 0 0 1 72
,
 1 0 0 0 0 1 0 0 0 0 46 0 0 0 46 27
,
 0 4 0 0 18 72 0 0 0 0 72 28 0 0 13 0
,
 72 69 0 0 0 1 0 0 0 0 72 2 0 0 59 1
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,46,46,0,0,0,27],[0,18,0,0,4,72,0,0,0,0,72,13,0,0,28,0],[72,0,0,0,69,1,0,0,0,0,72,59,0,0,2,1] >;

C2×U2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times {\rm U}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xU(2,3)");
// GroupNames label

G:=SmallGroup(192,981);
// by ID

G=gap.SmallGroup(192,981);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=b^2,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;
// generators/relations

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