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## G = C3×C8.A4order 288 = 25·32

### Direct product of C3 and C8.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×C8.A4
 Chief series C1 — C2 — Q8 — C4○D4 — C3×C4○D4 — C3×C4.A4 — C3×C8.A4
 Lower central Q8 — C3×C8.A4
 Upper central C1 — C24

Generators and relations for C3×C8.A4
G = < a,b,c,d,e | a3=b8=e3=1, c2=d2=b4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b4c, ece-1=b4cd, ede-1=c >

Subgroups: 144 in 58 conjugacy classes, 26 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C32, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C3×C6, C24, C24, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C8○D4, C3×C12, C2×C24, C3×M4(2), C4.A4, C3×C4○D4, C3×C24, C3×SL2(𝔽3), C8.A4, C3×C8○D4, C3×C4.A4, C3×C8.A4
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3×C6, C2×A4, C3×C12, C3×A4, C4×A4, C6×A4, C8.A4, C12×A4, C3×C8.A4

Smallest permutation representation of C3×C8.A4
On 96 points
Generators in S96
(1 37 29)(2 38 30)(3 39 31)(4 40 32)(5 33 25)(6 34 26)(7 35 27)(8 36 28)(9 95 87)(10 96 88)(11 89 81)(12 90 82)(13 91 83)(14 92 84)(15 93 85)(16 94 86)(17 75 67)(18 76 68)(19 77 69)(20 78 70)(21 79 71)(22 80 72)(23 73 65)(24 74 66)(41 59 51)(42 60 52)(43 61 53)(44 62 54)(45 63 55)(46 64 56)(47 57 49)(48 58 50)
(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)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)(65 87 69 83)(66 88 70 84)(67 81 71 85)(68 82 72 86)(73 95 77 91)(74 96 78 92)(75 89 79 93)(76 90 80 94)
(1 67 5 71)(2 68 6 72)(3 69 7 65)(4 70 8 66)(9 59 13 63)(10 60 14 64)(11 61 15 57)(12 62 16 58)(17 33 21 37)(18 34 22 38)(19 35 23 39)(20 36 24 40)(25 79 29 75)(26 80 30 76)(27 73 31 77)(28 74 32 78)(41 83 45 87)(42 84 46 88)(43 85 47 81)(44 86 48 82)(49 89 53 93)(50 90 54 94)(51 91 55 95)(52 92 56 96)
(1 37 29)(2 38 30)(3 39 31)(4 40 32)(5 33 25)(6 34 26)(7 35 27)(8 36 28)(9 51 65)(10 52 66)(11 53 67)(12 54 68)(13 55 69)(14 56 70)(15 49 71)(16 50 72)(17 89 43)(18 90 44)(19 91 45)(20 92 46)(21 93 47)(22 94 48)(23 95 41)(24 96 42)(57 79 85)(58 80 86)(59 73 87)(60 74 88)(61 75 81)(62 76 82)(63 77 83)(64 78 84)

G:=sub<Sym(96)| (1,37,29)(2,38,30)(3,39,31)(4,40,32)(5,33,25)(6,34,26)(7,35,27)(8,36,28)(9,95,87)(10,96,88)(11,89,81)(12,90,82)(13,91,83)(14,92,84)(15,93,85)(16,94,86)(17,75,67)(18,76,68)(19,77,69)(20,78,70)(21,79,71)(22,80,72)(23,73,65)(24,74,66)(41,59,51)(42,60,52)(43,61,53)(44,62,54)(45,63,55)(46,64,56)(47,57,49)(48,58,50), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64)(65,87,69,83)(66,88,70,84)(67,81,71,85)(68,82,72,86)(73,95,77,91)(74,96,78,92)(75,89,79,93)(76,90,80,94), (1,67,5,71)(2,68,6,72)(3,69,7,65)(4,70,8,66)(9,59,13,63)(10,60,14,64)(11,61,15,57)(12,62,16,58)(17,33,21,37)(18,34,22,38)(19,35,23,39)(20,36,24,40)(25,79,29,75)(26,80,30,76)(27,73,31,77)(28,74,32,78)(41,83,45,87)(42,84,46,88)(43,85,47,81)(44,86,48,82)(49,89,53,93)(50,90,54,94)(51,91,55,95)(52,92,56,96), (1,37,29)(2,38,30)(3,39,31)(4,40,32)(5,33,25)(6,34,26)(7,35,27)(8,36,28)(9,51,65)(10,52,66)(11,53,67)(12,54,68)(13,55,69)(14,56,70)(15,49,71)(16,50,72)(17,89,43)(18,90,44)(19,91,45)(20,92,46)(21,93,47)(22,94,48)(23,95,41)(24,96,42)(57,79,85)(58,80,86)(59,73,87)(60,74,88)(61,75,81)(62,76,82)(63,77,83)(64,78,84)>;

G:=Group( (1,37,29)(2,38,30)(3,39,31)(4,40,32)(5,33,25)(6,34,26)(7,35,27)(8,36,28)(9,95,87)(10,96,88)(11,89,81)(12,90,82)(13,91,83)(14,92,84)(15,93,85)(16,94,86)(17,75,67)(18,76,68)(19,77,69)(20,78,70)(21,79,71)(22,80,72)(23,73,65)(24,74,66)(41,59,51)(42,60,52)(43,61,53)(44,62,54)(45,63,55)(46,64,56)(47,57,49)(48,58,50), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64)(65,87,69,83)(66,88,70,84)(67,81,71,85)(68,82,72,86)(73,95,77,91)(74,96,78,92)(75,89,79,93)(76,90,80,94), (1,67,5,71)(2,68,6,72)(3,69,7,65)(4,70,8,66)(9,59,13,63)(10,60,14,64)(11,61,15,57)(12,62,16,58)(17,33,21,37)(18,34,22,38)(19,35,23,39)(20,36,24,40)(25,79,29,75)(26,80,30,76)(27,73,31,77)(28,74,32,78)(41,83,45,87)(42,84,46,88)(43,85,47,81)(44,86,48,82)(49,89,53,93)(50,90,54,94)(51,91,55,95)(52,92,56,96), (1,37,29)(2,38,30)(3,39,31)(4,40,32)(5,33,25)(6,34,26)(7,35,27)(8,36,28)(9,51,65)(10,52,66)(11,53,67)(12,54,68)(13,55,69)(14,56,70)(15,49,71)(16,50,72)(17,89,43)(18,90,44)(19,91,45)(20,92,46)(21,93,47)(22,94,48)(23,95,41)(24,96,42)(57,79,85)(58,80,86)(59,73,87)(60,74,88)(61,75,81)(62,76,82)(63,77,83)(64,78,84) );

G=PermutationGroup([[(1,37,29),(2,38,30),(3,39,31),(4,40,32),(5,33,25),(6,34,26),(7,35,27),(8,36,28),(9,95,87),(10,96,88),(11,89,81),(12,90,82),(13,91,83),(14,92,84),(15,93,85),(16,94,86),(17,75,67),(18,76,68),(19,77,69),(20,78,70),(21,79,71),(22,80,72),(23,73,65),(24,74,66),(41,59,51),(42,60,52),(43,61,53),(44,62,54),(45,63,55),(46,64,56),(47,57,49),(48,58,50)], [(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),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,43,5,47),(2,44,6,48),(3,45,7,41),(4,46,8,42),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(25,49,29,53),(26,50,30,54),(27,51,31,55),(28,52,32,56),(33,57,37,61),(34,58,38,62),(35,59,39,63),(36,60,40,64),(65,87,69,83),(66,88,70,84),(67,81,71,85),(68,82,72,86),(73,95,77,91),(74,96,78,92),(75,89,79,93),(76,90,80,94)], [(1,67,5,71),(2,68,6,72),(3,69,7,65),(4,70,8,66),(9,59,13,63),(10,60,14,64),(11,61,15,57),(12,62,16,58),(17,33,21,37),(18,34,22,38),(19,35,23,39),(20,36,24,40),(25,79,29,75),(26,80,30,76),(27,73,31,77),(28,74,32,78),(41,83,45,87),(42,84,46,88),(43,85,47,81),(44,86,48,82),(49,89,53,93),(50,90,54,94),(51,91,55,95),(52,92,56,96)], [(1,37,29),(2,38,30),(3,39,31),(4,40,32),(5,33,25),(6,34,26),(7,35,27),(8,36,28),(9,51,65),(10,52,66),(11,53,67),(12,54,68),(13,55,69),(14,56,70),(15,49,71),(16,50,72),(17,89,43),(18,90,44),(19,91,45),(20,92,46),(21,93,47),(22,94,48),(23,95,41),(24,96,42),(57,79,85),(58,80,86),(59,73,87),(60,74,88),(61,75,81),(62,76,82),(63,77,83),(64,78,84)]])

84 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3H 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E ··· 12P 12Q 12R 24A ··· 24H 24I ··· 24AF 24AG 24AH 24AI 24AJ order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 6 6 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 24 ··· 24 24 ··· 24 24 24 24 24 size 1 1 6 1 1 4 ··· 4 1 1 6 1 1 4 ··· 4 6 6 1 1 1 1 6 6 1 1 1 1 4 ··· 4 6 6 1 ··· 1 4 ··· 4 6 6 6 6

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 C8.A4 C3×C8.A4 A4 C2×A4 C3×A4 C4×A4 C6×A4 C12×A4 kernel C3×C8.A4 C3×C4.A4 C8.A4 C3×C8○D4 C3×SL2(𝔽3) C4.A4 C3×C4○D4 SL2(𝔽3) C3×Q8 C3 C1 C24 C12 C8 C6 C4 C2 # reps 1 1 6 2 2 6 2 12 4 12 24 1 1 2 2 2 4

Matrix representation of C3×C8.A4 in GL2(𝔽73) generated by

 64 0 0 64
,
 51 0 0 51
,
 9 65 65 64
,
 0 1 72 0
,
 64 72 0 8
G:=sub<GL(2,GF(73))| [64,0,0,64],[51,0,0,51],[9,65,65,64],[0,72,1,0],[64,0,72,8] >;

C3×C8.A4 in GAP, Magma, Sage, TeX

C_3\times C_8.A_4
% in TeX

G:=Group("C3xC8.A4");
// GroupNames label

G:=SmallGroup(288,638);
// by ID

G=gap.SmallGroup(288,638);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,2,-2,126,248,1271,172,2280,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=e^3=1,c^2=d^2=b^4,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^4*c,e*c*e^-1=b^4*c*d,e*d*e^-1=c>;
// generators/relations

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