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## G = C22×CSU2(𝔽3)  order 192 = 26·3

### Direct product of C22 and CSU2(𝔽3)

Aliases: C22×CSU2(𝔽3), C23.20S4, SL2(𝔽3).1C23, C2.8(C22×S4), (C2×Q8).20D6, C22.25(C2×S4), Q8.1(C22×S3), (C22×Q8).5S3, (C22×SL2(𝔽3)).5C2, (C2×SL2(𝔽3)).20C22, SmallGroup(192,1474)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C22×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — CSU2(𝔽3) — C2×CSU2(𝔽3) — C22×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C22×CSU2(𝔽3)
 Upper central C1 — C23

Generators and relations for C22×CSU2(𝔽3)
G = < a,b,c,d,e,f | a2=b2=c4=e3=1, d2=f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf-1=c-1, ece-1=cd, fcf-1=c2d, ede-1=c, fef-1=e-1 >

Subgroups: 475 in 153 conjugacy classes, 37 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C8, C2×C4, Q8, Q8, C23, Dic3, C2×C6, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C22×C6, C22×C8, C2×Q16, C22×Q8, C22×Q8, CSU2(𝔽3), C2×SL2(𝔽3), C22×Dic3, C22×Q16, C2×CSU2(𝔽3), C22×SL2(𝔽3), C22×CSU2(𝔽3)
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, CSU2(𝔽3), C2×S4, C2×CSU2(𝔽3), C22×S4, C22×CSU2(𝔽3)

Smallest permutation representation of C22×CSU2(𝔽3)
On 64 points
Generators in S64
(1 24)(2 21)(3 22)(4 23)(5 49)(6 50)(7 51)(8 52)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 35)(18 36)(19 33)(20 34)(37 55)(38 56)(39 53)(40 54)(41 59)(42 60)(43 57)(44 58)(45 63)(46 64)(47 61)(48 62)
(1 16)(2 13)(3 14)(4 15)(5 57)(6 58)(7 59)(8 60)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)(25 35)(26 36)(27 33)(28 34)(37 47)(38 48)(39 45)(40 46)(41 51)(42 52)(43 49)(44 50)(53 63)(54 64)(55 61)(56 62)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 11 3 9)(2 10 4 12)(5 63 7 61)(6 62 8 64)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)(29 36 31 34)(30 35 32 33)(37 43 39 41)(38 42 40 44)(45 51 47 49)(46 50 48 52)(53 59 55 57)(54 58 56 60)
(2 11 10)(4 9 12)(5 8 62)(6 64 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)(29 33 36)(31 35 34)(38 43 42)(40 41 44)(46 51 50)(48 49 52)(54 59 58)(56 57 60)
(1 61 3 63)(2 7 4 5)(6 12 8 10)(9 62 11 64)(13 59 15 57)(14 53 16 55)(17 54 19 56)(18 60 20 58)(21 51 23 49)(22 45 24 47)(25 46 27 48)(26 52 28 50)(29 43 31 41)(30 37 32 39)(33 38 35 40)(34 44 36 42)

G:=sub<Sym(64)| (1,24)(2,21)(3,22)(4,23)(5,49)(6,50)(7,51)(8,52)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,35)(18,36)(19,33)(20,34)(37,55)(38,56)(39,53)(40,54)(41,59)(42,60)(43,57)(44,58)(45,63)(46,64)(47,61)(48,62), (1,16)(2,13)(3,14)(4,15)(5,57)(6,58)(7,59)(8,60)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30)(25,35)(26,36)(27,33)(28,34)(37,47)(38,48)(39,45)(40,46)(41,51)(42,52)(43,49)(44,50)(53,63)(54,64)(55,61)(56,62), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,11,3,9)(2,10,4,12)(5,63,7,61)(6,62,8,64)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(29,36,31,34)(30,35,32,33)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60), (2,11,10)(4,9,12)(5,8,62)(6,64,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(29,33,36)(31,35,34)(38,43,42)(40,41,44)(46,51,50)(48,49,52)(54,59,58)(56,57,60), (1,61,3,63)(2,7,4,5)(6,12,8,10)(9,62,11,64)(13,59,15,57)(14,53,16,55)(17,54,19,56)(18,60,20,58)(21,51,23,49)(22,45,24,47)(25,46,27,48)(26,52,28,50)(29,43,31,41)(30,37,32,39)(33,38,35,40)(34,44,36,42)>;

G:=Group( (1,24)(2,21)(3,22)(4,23)(5,49)(6,50)(7,51)(8,52)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,35)(18,36)(19,33)(20,34)(37,55)(38,56)(39,53)(40,54)(41,59)(42,60)(43,57)(44,58)(45,63)(46,64)(47,61)(48,62), (1,16)(2,13)(3,14)(4,15)(5,57)(6,58)(7,59)(8,60)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30)(25,35)(26,36)(27,33)(28,34)(37,47)(38,48)(39,45)(40,46)(41,51)(42,52)(43,49)(44,50)(53,63)(54,64)(55,61)(56,62), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,11,3,9)(2,10,4,12)(5,63,7,61)(6,62,8,64)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(29,36,31,34)(30,35,32,33)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60), (2,11,10)(4,9,12)(5,8,62)(6,64,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(29,33,36)(31,35,34)(38,43,42)(40,41,44)(46,51,50)(48,49,52)(54,59,58)(56,57,60), (1,61,3,63)(2,7,4,5)(6,12,8,10)(9,62,11,64)(13,59,15,57)(14,53,16,55)(17,54,19,56)(18,60,20,58)(21,51,23,49)(22,45,24,47)(25,46,27,48)(26,52,28,50)(29,43,31,41)(30,37,32,39)(33,38,35,40)(34,44,36,42) );

G=PermutationGroup([[(1,24),(2,21),(3,22),(4,23),(5,49),(6,50),(7,51),(8,52),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30),(17,35),(18,36),(19,33),(20,34),(37,55),(38,56),(39,53),(40,54),(41,59),(42,60),(43,57),(44,58),(45,63),(46,64),(47,61),(48,62)], [(1,16),(2,13),(3,14),(4,15),(5,57),(6,58),(7,59),(8,60),(9,19),(10,20),(11,17),(12,18),(21,31),(22,32),(23,29),(24,30),(25,35),(26,36),(27,33),(28,34),(37,47),(38,48),(39,45),(40,46),(41,51),(42,52),(43,49),(44,50),(53,63),(54,64),(55,61),(56,62)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,11,3,9),(2,10,4,12),(5,63,7,61),(6,62,8,64),(13,20,15,18),(14,19,16,17),(21,28,23,26),(22,27,24,25),(29,36,31,34),(30,35,32,33),(37,43,39,41),(38,42,40,44),(45,51,47,49),(46,50,48,52),(53,59,55,57),(54,58,56,60)], [(2,11,10),(4,9,12),(5,8,62),(6,64,7),(13,17,20),(15,19,18),(21,25,28),(23,27,26),(29,33,36),(31,35,34),(38,43,42),(40,41,44),(46,51,50),(48,49,52),(54,59,58),(56,57,60)], [(1,61,3,63),(2,7,4,5),(6,12,8,10),(9,62,11,64),(13,59,15,57),(14,53,16,55),(17,54,19,56),(18,60,20,58),(21,51,23,49),(22,45,24,47),(25,46,27,48),(26,52,28,50),(29,43,31,41),(30,37,32,39),(33,38,35,40),(34,44,36,42)]])

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 2 2 2 3 3 4 type + + + + + - + + - image C1 C2 C2 S3 D6 CSU2(𝔽3) S4 C2×S4 CSU2(𝔽3) kernel C22×CSU2(𝔽3) C2×CSU2(𝔽3) C22×SL2(𝔽3) C22×Q8 C2×Q8 C22 C23 C22 C22 # reps 1 6 1 1 3 8 2 6 4

Matrix representation of C22×CSU2(𝔽3) in GL6(𝔽73)

 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 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 62 12 0 0 0 0 2 11
,
 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 13 71 0 0 0 0 12 60
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 54 5 0 0 0 0 59 19

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,62,2,0,0,0,0,12,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,12,0,0,0,0,71,60],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,54,59,0,0,0,0,5,19] >;

C22×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_2^2\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C2^2xCSU(2,3)");
// GroupNames label

G:=SmallGroup(192,1474);
// by ID

G=gap.SmallGroup(192,1474);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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