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## G = C2×A4⋊C8order 192 = 26·3

### Direct product of C2 and A4⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×A4⋊C8
G = < a,b,c,d,e | a2=b2=c2=d3=e8=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: 254 in 89 conjugacy classes, 27 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, C23, C23, C23, C12, A4, C2×C6, C2×C8, C22×C4, C22×C4, C24, C3⋊C8, C2×C12, C2×A4, C2×A4, C22⋊C8, C22×C8, C23×C4, C2×C3⋊C8, C4×A4, C22×A4, C2×C22⋊C8, A4⋊C8, C2×C4×A4, C2×A4⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, C2×Dic3, S4, C2×C3⋊C8, A4⋊C4, C2×S4, A4⋊C8, C2×A4⋊C4, C2×A4⋊C8

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

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

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

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

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 type + + + + - + - + + image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 Dic3 C3⋊C8 S4 A4⋊C4 C2×S4 A4⋊C4 A4⋊C8 kernel C2×A4⋊C8 A4⋊C8 C2×C4×A4 C4×A4 C22×A4 C2×A4 C23×C4 C22×C4 C22×C4 C24 C23 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 8 1 1 1 1 4 2 2 2 2 8

Matrix representation of C2×A4⋊C8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 72
,
 0 72 0 0 0 1 72 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 29 42 0 0 0 71 44 0 0 0 0 0 0 0 72 0 0 0 72 0 0 0 72 0 0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[0,1,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[29,71,0,0,0,42,44,0,0,0,0,0,0,0,72,0,0,0,72,0,0,0,72,0,0] >;

C2×A4⋊C8 in GAP, Magma, Sage, TeX

C_2\times A_4\rtimes C_8
% in TeX

G:=Group("C2xA4:C8");
// GroupNames label

G:=SmallGroup(192,967);
// by ID

G=gap.SmallGroup(192,967);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,58,1124,4037,285,2358,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^3=e^8=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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