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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 [×3], C4, C4, C22, C6, C6 [×4], C8, C8, C2×C4, D4, Q8, C32, C12, C12 [×4], C2×C6, C2×C8, M4(2), C4○D4, C3×C6, C24, C24 [×4], SL2(𝔽3) [×3], C2×C12, C3×D4, C3×Q8, C8○D4, C3×C12, C2×C24, C3×M4(2), C4.A4 [×3], C3×C4○D4, C3×C24, C3×SL2(𝔽3), C8.A4 [×3], C3×C8○D4, C3×C4.A4, C3×C8.A4
Quotients: C1, C2, C3 [×4], C4, C6 [×4], C32, C12 [×4], 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 80 23)(2 73 24)(3 74 17)(4 75 18)(5 76 19)(6 77 20)(7 78 21)(8 79 22)(9 47 57)(10 48 58)(11 41 59)(12 42 60)(13 43 61)(14 44 62)(15 45 63)(16 46 64)(25 65 81)(26 66 82)(27 67 83)(28 68 84)(29 69 85)(30 70 86)(31 71 87)(32 72 88)(33 51 90)(34 52 91)(35 53 92)(36 54 93)(37 55 94)(38 56 95)(39 49 96)(40 50 89)
(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 23 13 19)(10 24 14 20)(11 17 15 21)(12 18 16 22)(25 93 29 89)(26 94 30 90)(27 95 31 91)(28 96 32 92)(33 66 37 70)(34 67 38 71)(35 68 39 72)(36 69 40 65)(49 88 53 84)(50 81 54 85)(51 82 55 86)(52 83 56 87)(57 80 61 76)(58 73 62 77)(59 74 63 78)(60 75 64 79)
(1 65 5 69)(2 66 6 70)(3 67 7 71)(4 68 8 72)(9 93 13 89)(10 94 14 90)(11 95 15 91)(12 96 16 92)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)(33 48 37 44)(34 41 38 45)(35 42 39 46)(36 43 40 47)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)(73 82 77 86)(74 83 78 87)(75 84 79 88)(76 85 80 81)
(1 80 23)(2 73 24)(3 74 17)(4 75 18)(5 76 19)(6 77 20)(7 78 21)(8 79 22)(9 69 50)(10 70 51)(11 71 52)(12 72 53)(13 65 54)(14 66 55)(15 67 56)(16 68 49)(25 36 61)(26 37 62)(27 38 63)(28 39 64)(29 40 57)(30 33 58)(31 34 59)(32 35 60)(41 87 91)(42 88 92)(43 81 93)(44 82 94)(45 83 95)(46 84 96)(47 85 89)(48 86 90)

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

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

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

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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