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## G = C4×CSU2(𝔽3)  order 192 = 26·3

### Direct product of C4 and CSU2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C4×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×CSU2(𝔽3) — C4×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C4×CSU2(𝔽3)
 Upper central C1 — C2×C4

Generators and relations for C4×CSU2(𝔽3)
G = < a,b,c,d,e | a4=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

Subgroups: 223 in 70 conjugacy classes, 19 normal (15 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C4×Q8, C2×Q16, C4×Dic3, CSU2(𝔽3), C2×SL2(𝔽3), C4×Q16, Q8⋊Dic3, C4×SL2(𝔽3), C2×CSU2(𝔽3), C4×CSU2(𝔽3)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, CSU2(𝔽3), C2×S4, C4×S4, C2×CSU2(𝔽3), C4.6S4, C4×CSU2(𝔽3)

Smallest permutation representation of C4×CSU2(𝔽3)
On 64 points
Generators in S64
(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 34 5 29)(2 35 6 30)(3 36 7 31)(4 33 8 32)(9 18 55 38)(10 19 56 39)(11 20 53 40)(12 17 54 37)(13 22 42 51)(14 23 43 52)(15 24 44 49)(16 21 41 50)(25 60 45 61)(26 57 46 62)(27 58 47 63)(28 59 48 64)
(1 51 5 22)(2 52 6 23)(3 49 7 24)(4 50 8 21)(9 26 55 46)(10 27 56 47)(11 28 53 48)(12 25 54 45)(13 34 42 29)(14 35 43 30)(15 36 44 31)(16 33 41 32)(17 61 37 60)(18 62 38 57)(19 63 39 58)(20 64 40 59)
(9 57 26)(10 58 27)(11 59 28)(12 60 25)(13 29 22)(14 30 23)(15 31 24)(16 32 21)(33 50 41)(34 51 42)(35 52 43)(36 49 44)(45 54 61)(46 55 62)(47 56 63)(48 53 64)
(1 37 5 17)(2 38 6 18)(3 39 7 19)(4 40 8 20)(9 52 55 23)(10 49 56 24)(11 50 53 21)(12 51 54 22)(13 25 42 45)(14 26 43 46)(15 27 44 47)(16 28 41 48)(29 60 34 61)(30 57 35 62)(31 58 36 63)(32 59 33 64)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C4 S3 D6 C4×S3 CSU2(𝔽3) C4.6S4 S4 C2×S4 C4×S4 CSU2(𝔽3) C4.6S4 kernel C4×CSU2(𝔽3) Q8⋊Dic3 C4×SL2(𝔽3) C2×CSU2(𝔽3) CSU2(𝔽3) C4×Q8 C2×Q8 Q8 C4 C2 C2×C4 C22 C2 C4 C2 # reps 1 1 1 1 4 1 1 2 4 4 2 2 4 2 2

Matrix representation of C4×CSU2(𝔽3) in GL4(𝔽73) generated by

 46 0 0 0 0 46 0 0 0 0 46 0 0 0 0 46
,
 1 0 0 0 0 1 0 0 0 0 45 56 0 0 29 28
,
 1 0 0 0 0 1 0 0 0 0 57 44 0 0 29 16
,
 0 72 0 0 1 72 0 0 0 0 0 72 0 0 1 72
,
 0 72 0 0 72 0 0 0 0 0 27 46 0 0 0 46
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,45,29,0,0,56,28],[1,0,0,0,0,1,0,0,0,0,57,29,0,0,44,16],[0,1,0,0,72,72,0,0,0,0,0,1,0,0,72,72],[0,72,0,0,72,0,0,0,0,0,27,0,0,0,46,46] >;

C4×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_4\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C4xCSU(2,3)");
// GroupNames label

G:=SmallGroup(192,946);
// by ID

G=gap.SmallGroup(192,946);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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