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## G = C4×A4⋊C4order 192 = 26·3

### Direct product of C4 and A4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C4×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — C22×A4 — C2×A4⋊C4 — C4×A4⋊C4
 Lower central A4 — C4×A4⋊C4
 Upper central C1 — C2×C4

Generators and relations for C4×A4⋊C4
G = < a,b,c,d,e | a4=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 394 in 125 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, A4, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×A4, C2×A4, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×Dic3, A4⋊C4, C4×A4, C22×A4, C4×C22⋊C4, C2×A4⋊C4, C2×C4×A4, C4×A4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, S4, C4×Dic3, A4⋊C4, C2×S4, C4×S4, C2×A4⋊C4, C4×A4⋊C4

Smallest permutation representation of C4×A4⋊C4
On 48 points
Generators in S48
(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)
(1 31)(2 32)(3 29)(4 30)(5 48)(6 45)(7 46)(8 47)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)(25 43)(26 44)(27 41)(28 42)(33 38)(34 39)(35 40)(36 37)
(1 24)(2 21)(3 22)(4 23)(5 28)(6 25)(7 26)(8 27)(9 35)(10 36)(11 33)(12 34)(13 40)(14 37)(15 38)(16 39)(17 30)(18 31)(19 32)(20 29)(41 47)(42 48)(43 45)(44 46)
(1 27 39)(2 28 40)(3 25 37)(4 26 38)(5 9 32)(6 10 29)(7 11 30)(8 12 31)(13 19 42)(14 20 43)(15 17 44)(16 18 41)(21 48 35)(22 45 36)(23 46 33)(24 47 34)
(1 29 24 20)(2 30 21 17)(3 31 22 18)(4 32 23 19)(5 33 42 38)(6 34 43 39)(7 35 44 40)(8 36 41 37)(9 46 13 26)(10 47 14 27)(11 48 15 28)(12 45 16 25)

G:=sub<Sym(48)| (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), (1,31)(2,32)(3,29)(4,30)(5,48)(6,45)(7,46)(8,47)(9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21)(20,22)(25,43)(26,44)(27,41)(28,42)(33,38)(34,39)(35,40)(36,37), (1,24)(2,21)(3,22)(4,23)(5,28)(6,25)(7,26)(8,27)(9,35)(10,36)(11,33)(12,34)(13,40)(14,37)(15,38)(16,39)(17,30)(18,31)(19,32)(20,29)(41,47)(42,48)(43,45)(44,46), (1,27,39)(2,28,40)(3,25,37)(4,26,38)(5,9,32)(6,10,29)(7,11,30)(8,12,31)(13,19,42)(14,20,43)(15,17,44)(16,18,41)(21,48,35)(22,45,36)(23,46,33)(24,47,34), (1,29,24,20)(2,30,21,17)(3,31,22,18)(4,32,23,19)(5,33,42,38)(6,34,43,39)(7,35,44,40)(8,36,41,37)(9,46,13,26)(10,47,14,27)(11,48,15,28)(12,45,16,25)>;

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), (1,31)(2,32)(3,29)(4,30)(5,48)(6,45)(7,46)(8,47)(9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,21)(20,22)(25,43)(26,44)(27,41)(28,42)(33,38)(34,39)(35,40)(36,37), (1,24)(2,21)(3,22)(4,23)(5,28)(6,25)(7,26)(8,27)(9,35)(10,36)(11,33)(12,34)(13,40)(14,37)(15,38)(16,39)(17,30)(18,31)(19,32)(20,29)(41,47)(42,48)(43,45)(44,46), (1,27,39)(2,28,40)(3,25,37)(4,26,38)(5,9,32)(6,10,29)(7,11,30)(8,12,31)(13,19,42)(14,20,43)(15,17,44)(16,18,41)(21,48,35)(22,45,36)(23,46,33)(24,47,34), (1,29,24,20)(2,30,21,17)(3,31,22,18)(4,32,23,19)(5,33,42,38)(6,34,43,39)(7,35,44,40)(8,36,41,37)(9,46,13,26)(10,47,14,27)(11,48,15,28)(12,45,16,25) );

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)], [(1,31),(2,32),(3,29),(4,30),(5,48),(6,45),(7,46),(8,47),(9,13),(10,14),(11,15),(12,16),(17,23),(18,24),(19,21),(20,22),(25,43),(26,44),(27,41),(28,42),(33,38),(34,39),(35,40),(36,37)], [(1,24),(2,21),(3,22),(4,23),(5,28),(6,25),(7,26),(8,27),(9,35),(10,36),(11,33),(12,34),(13,40),(14,37),(15,38),(16,39),(17,30),(18,31),(19,32),(20,29),(41,47),(42,48),(43,45),(44,46)], [(1,27,39),(2,28,40),(3,25,37),(4,26,38),(5,9,32),(6,10,29),(7,11,30),(8,12,31),(13,19,42),(14,20,43),(15,17,44),(16,18,41),(21,48,35),(22,45,36),(23,46,33),(24,47,34)], [(1,29,24,20),(2,30,21,17),(3,31,22,18),(4,32,23,19),(5,33,42,38),(6,34,43,39),(7,35,44,40),(8,36,41,37),(9,46,13,26),(10,47,14,27),(11,48,15,28),(12,45,16,25)]])

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + - + + + image C1 C2 C2 C4 C4 S3 Dic3 D6 C4×S3 S4 A4⋊C4 C2×S4 C4×S4 kernel C4×A4⋊C4 C2×A4⋊C4 C2×C4×A4 A4⋊C4 C4×A4 C23×C4 C22×C4 C24 C23 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 1 2 1 4 2 4 2 8

Matrix representation of C4×A4⋊C4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 0 12 0 0 0 1 12 0 0 0 0 0 0 8 0 0 0 0 0 5 0 0 1 0 0
,
 0 8 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 8

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

C4×A4⋊C4 in GAP, Magma, Sage, TeX

C_4\times A_4\rtimes C_4
% in TeX

G:=Group("C4xA4:C4");
// GroupNames label

G:=SmallGroup(192,969);
// by ID

G=gap.SmallGroup(192,969);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,64,1124,4037,285,2358,475]);
// Polycyclic

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

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