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## G = C22×C4.A4order 192 = 26·3

### Direct product of C22 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C22×C4.A4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C22×SL2(𝔽3) — C22×C4.A4
 Lower central Q8 — C22×C4.A4
 Upper central C1 — C22×C4

Generators and relations for C22×C4.A4
G = < a,b,c,d,e,f | a2=b2=c4=f3=1, d2=e2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=c2d, fdf-1=c2de, fef-1=d >

Subgroups: 629 in 220 conjugacy classes, 59 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C12, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C2×C12, C22×C6, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×SL2(𝔽3), C4.A4, C22×C12, C22×C4○D4, C22×SL2(𝔽3), C2×C4.A4, C22×C4.A4
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, C2×A4, C22×C6, C4.A4, C22×A4, C2×C4.A4, C23×A4, C22×C4.A4

Smallest permutation representation of C22×C4.A4
On 64 points
Generators in S64
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)(17 29)(18 30)(19 31)(20 32)(21 42)(22 43)(23 44)(24 41)(25 39)(26 40)(27 37)(28 38)(33 47)(34 48)(35 45)(36 46)(49 56)(50 53)(51 54)(52 55)(57 64)(58 61)(59 62)(60 63)
(1 37)(2 38)(3 39)(4 40)(5 34)(6 35)(7 36)(8 33)(9 45)(10 46)(11 47)(12 48)(13 27)(14 28)(15 25)(16 26)(17 41)(18 42)(19 43)(20 44)(21 30)(22 31)(23 32)(24 29)(49 59)(50 60)(51 57)(52 58)(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 53 3 55)(2 54 4 56)(5 21 7 23)(6 22 8 24)(9 43 11 41)(10 44 12 42)(13 50 15 52)(14 51 16 49)(17 45 19 47)(18 46 20 48)(25 58 27 60)(26 59 28 57)(29 35 31 33)(30 36 32 34)(37 63 39 61)(38 64 40 62)
(1 47 3 45)(2 48 4 46)(5 26 7 28)(6 27 8 25)(9 37 11 39)(10 38 12 40)(13 33 15 35)(14 34 16 36)(17 53 19 55)(18 54 20 56)(21 57 23 59)(22 58 24 60)(29 50 31 52)(30 51 32 49)(41 63 43 61)(42 64 44 62)
(5 21 57)(6 22 58)(7 23 59)(8 24 60)(9 43 61)(10 44 62)(11 41 63)(12 42 64)(17 53 47)(18 54 48)(19 55 45)(20 56 46)(29 50 33)(30 51 34)(31 52 35)(32 49 36)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 3 3 3 type + + + + + + image C1 C2 C2 C3 C6 C6 C4.A4 A4 C2×A4 C2×A4 kernel C22×C4.A4 C22×SL2(𝔽3) C2×C4.A4 C22×C4○D4 C22×Q8 C2×C4○D4 C22 C22×C4 C2×C4 C23 # reps 1 1 6 2 2 12 24 1 6 1

Matrix representation of C22×C4.A4 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 12 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 12 12 0 0 2 1
,
 1 0 0 0 0 1 0 0 0 0 7 3 0 0 5 6
,
 3 0 0 0 0 9 0 0 0 0 1 4 0 0 0 9
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,2,0,0,12,1],[1,0,0,0,0,1,0,0,0,0,7,5,0,0,3,6],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,4,9] >;

C22×C4.A4 in GAP, Magma, Sage, TeX

C_2^2\times C_4.A_4
% in TeX

G:=Group("C2^2xC4.A4");
// GroupNames label

G:=SmallGroup(192,1500);
// by ID

G=gap.SmallGroup(192,1500);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,520,235,172,404,285,124]);
// Polycyclic

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

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