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## G = C2×C4×SL2(𝔽3)  order 192 = 26·3

### Direct product of C2×C4 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×C4×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C22×SL2(𝔽3) — C2×C4×SL2(𝔽3)
 Lower central Q8 — C2×C4×SL2(𝔽3)
 Upper central C1 — C22×C4

Generators and relations for C2×C4×SL2(𝔽3)
G = < a,b,c,d,e | a2=b4=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 277 in 108 conjugacy classes, 43 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, Q8, Q8, C23, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, SL2(𝔽3), C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C22×Q8, C2×SL2(𝔽3), C2×SL2(𝔽3), C22×C12, C2×C4×Q8, C4×SL2(𝔽3), C22×SL2(𝔽3), C2×C4×SL2(𝔽3)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, SL2(𝔽3), C2×C12, C2×A4, C4×A4, C2×SL2(𝔽3), C4.A4, C22×A4, C4×SL2(𝔽3), C2×C4×A4, C22×SL2(𝔽3), C2×C4.A4, C2×C4×SL2(𝔽3)

Smallest permutation representation of C2×C4×SL2(𝔽3)
On 64 points
Generators in S64
(1 20)(2 17)(3 18)(4 19)(5 62)(6 63)(7 64)(8 61)(9 16)(10 13)(11 14)(12 15)(21 28)(22 25)(23 26)(24 27)(29 57)(30 58)(31 59)(32 60)(33 37)(34 38)(35 39)(36 40)(41 55)(42 56)(43 53)(44 54)(45 49)(46 50)(47 51)(48 52)
(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 15 35 51)(2 16 36 52)(3 13 33 49)(4 14 34 50)(5 44 30 22)(6 41 31 23)(7 42 32 24)(8 43 29 21)(9 40 48 17)(10 37 45 18)(11 38 46 19)(12 39 47 20)(25 62 54 58)(26 63 55 59)(27 64 56 60)(28 61 53 57)
(1 8 35 29)(2 5 36 30)(3 6 33 31)(4 7 34 32)(9 25 48 54)(10 26 45 55)(11 27 46 56)(12 28 47 53)(13 23 49 41)(14 24 50 42)(15 21 51 43)(16 22 52 44)(17 62 40 58)(18 63 37 59)(19 64 38 60)(20 61 39 57)
(5 16 22)(6 13 23)(7 14 24)(8 15 21)(9 25 62)(10 26 63)(11 27 64)(12 28 61)(29 51 43)(30 52 44)(31 49 41)(32 50 42)(45 55 59)(46 56 60)(47 53 57)(48 54 58)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 4I ··· 4P 6A ··· 6N 12A ··· 12P order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 4 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 type + + + - + + + image C1 C2 C2 C3 C4 C6 C6 C12 SL2(𝔽3) SL2(𝔽3) C4.A4 A4 C2×A4 C2×A4 C4×A4 kernel C2×C4×SL2(𝔽3) C4×SL2(𝔽3) C22×SL2(𝔽3) C2×C4×Q8 C2×SL2(𝔽3) C4×Q8 C22×Q8 C2×Q8 C2×C4 C2×C4 C22 C22×C4 C2×C4 C23 C22 # reps 1 2 1 2 4 4 2 8 4 8 12 1 2 1 4

Matrix representation of C2×C4×SL2(𝔽3) in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 8 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 4 3 0 0 3 9
,
 3 0 0 0 0 9 0 0 0 0 1 3 0 0 0 9
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,4,3,0,0,3,9],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,3,9] >;

C2×C4×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xC4xSL(2,3)");
// GroupNames label

G:=SmallGroup(192,996);
// by ID

G=gap.SmallGroup(192,996);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,92,438,172,775,285,124]);
// Polycyclic

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

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