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## G = A4×C2×C8order 192 = 26·3

### Direct product of C2×C8 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C2×C8
 Chief series C1 — C22 — C23 — C22×C4 — C4×A4 — C2×C4×A4 — A4×C2×C8
 Lower central C22 — A4×C2×C8
 Upper central C1 — C2×C8

Generators and relations for A4×C2×C8
G = < a,b,c,d,e | a2=b8=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 224 in 93 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, C2×C4, C23, C23, C23, C12, A4, C2×C6, C2×C8, C2×C8, C22×C4, C22×C4, C24, C24, C2×C12, C2×A4, C2×A4, C22×C8, C22×C8, C23×C4, C2×C24, C4×A4, C22×A4, C23×C8, C8×A4, C2×C4×A4, A4×C2×C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, A4, C2×C6, C2×C8, C24, C2×C12, C2×A4, C2×C24, C4×A4, C22×A4, C8×A4, C2×C4×A4, A4×C2×C8

Smallest permutation representation of A4×C2×C8
On 48 points
Generators in S48
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 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 5)(2 6)(3 7)(4 8)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 21)(18 22)(19 23)(20 24)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 15 39)(2 16 40)(3 9 33)(4 10 34)(5 11 35)(6 12 36)(7 13 37)(8 14 38)(17 32 41)(18 25 42)(19 26 43)(20 27 44)(21 28 45)(22 29 46)(23 30 47)(24 31 48)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 8A ··· 8H 8I ··· 8P 12A ··· 12H 24A ··· 24P order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 3 3 3 3 4 4 1 1 1 1 3 3 3 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 A4 C2×A4 C2×A4 C4×A4 C4×A4 C8×A4 kernel A4×C2×C8 C8×A4 C2×C4×A4 C23×C8 C4×A4 C22×A4 C22×C8 C23×C4 C2×A4 C22×C4 C24 C23 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 2 1 2 2 8

Matrix representation of A4×C2×C8 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 72 0 0 0 0 72 0 0 0 0 1
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C2×C8 in GAP, Magma, Sage, TeX

A_4\times C_2\times C_8
% in TeX

G:=Group("A4xC2xC8");
// GroupNames label

G:=SmallGroup(192,1010);
// by ID

G=gap.SmallGroup(192,1010);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,92,80,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^2=e^3=1,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,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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