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## G = C4×C4.A4order 192 = 26·3

### Direct product of C4 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4×C4.A4
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C4×SL2(𝔽3) — C4×C4.A4
 Lower central Q8 — C4×C4.A4
 Upper central C1 — C42

Generators and relations for C4×C4.A4
G = < a,b,c,d,e | a4=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Subgroups: 253 in 90 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×C12, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×C12, C2×SL2(𝔽3), C4.A4, C4×C4○D4, C4×SL2(𝔽3), C2×C4.A4, C4×C4.A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C2×C12, C2×A4, C4×A4, C4.A4, C22×A4, C2×C4×A4, C2×C4.A4, C4×C4.A4

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A ··· 4L 4M ··· 4R 6A ··· 6F 12A ··· 12X order 1 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 6 6 4 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 3 3 3 type + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 C4.A4 A4 C2×A4 C4×A4 kernel C4×C4.A4 C4×SL2(𝔽3) C2×C4.A4 C4×C4○D4 C4.A4 C4×Q8 C2×C4○D4 C4○D4 C4 C42 C2×C4 C4 # reps 1 2 1 2 4 4 2 8 24 1 3 4

Matrix representation of C4×C4.A4 in GL3(𝔽13) generated by

 8 0 0 0 8 0 0 0 8
,
 1 0 0 0 5 0 0 0 5
,
 1 0 0 0 10 4 0 4 3
,
 1 0 0 0 0 1 0 12 0
,
 9 0 0 0 1 4 0 0 3
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,5,0,0,0,5],[1,0,0,0,10,4,0,4,3],[1,0,0,0,0,12,0,1,0],[9,0,0,0,1,0,0,4,3] >;

C4×C4.A4 in GAP, Magma, Sage, TeX

C_4\times C_4.A_4
% in TeX

G:=Group("C4xC4.A4");
// GroupNames label

G:=SmallGroup(192,997);
// by ID

G=gap.SmallGroup(192,997);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,84,680,438,172,775,285,124]);
// Polycyclic

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

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