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## G = C8⋊(C4×S3)  order 192 = 26·3

### 3rd semidirect product of C8 and C4×S3 acting via C4×S3/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C8⋊(C4×S3)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×C8⋊S3 — C8⋊(C4×S3)
 Lower central C3 — C6 — C12 — C8⋊(C4×S3)
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8⋊(C4×S3)
G = < a,b,c,d | a8=b4=c3=d2=1, bab-1=a3, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 304 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.Q8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, M4(2)⋊C4, C6.Q16, C12.Q8, C241C4, C3×C4.Q8, S3×C4⋊C4, C4⋊C47S3, C2×C8⋊S3, C8⋊(C4×S3)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4, S3×Q8, M4(2)⋊C4, S3×C4⋊C4, Q83D6, D4.D6, C8⋊(C4×S3)

Smallest permutation representation of C8⋊(C4×S3)
On 96 points
Generators in S96
(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 15 45 79)(2 10 46 74)(3 13 47 77)(4 16 48 80)(5 11 41 75)(6 14 42 78)(7 9 43 73)(8 12 44 76)(17 61 95 36)(18 64 96 39)(19 59 89 34)(20 62 90 37)(21 57 91 40)(22 60 92 35)(23 63 93 38)(24 58 94 33)(25 52 84 65)(26 55 85 68)(27 50 86 71)(28 53 87 66)(29 56 88 69)(30 51 81 72)(31 54 82 67)(32 49 83 70)
(1 17 82)(2 18 83)(3 19 84)(4 20 85)(5 21 86)(6 22 87)(7 23 88)(8 24 81)(9 63 69)(10 64 70)(11 57 71)(12 58 72)(13 59 65)(14 60 66)(15 61 67)(16 62 68)(25 47 89)(26 48 90)(27 41 91)(28 42 92)(29 43 93)(30 44 94)(31 45 95)(32 46 96)(33 51 76)(34 52 77)(35 53 78)(36 54 79)(37 55 80)(38 56 73)(39 49 74)(40 50 75)
(2 6)(4 8)(10 14)(12 16)(17 82)(18 87)(19 84)(20 81)(21 86)(22 83)(23 88)(24 85)(25 89)(26 94)(27 91)(28 96)(29 93)(30 90)(31 95)(32 92)(33 55)(34 52)(35 49)(36 54)(37 51)(38 56)(39 53)(40 50)(42 46)(44 48)(57 71)(58 68)(59 65)(60 70)(61 67)(62 72)(63 69)(64 66)(74 78)(76 80)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - + + + + + - - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 Q8 D4 D4 D6 D6 C4×S3 C8⋊C22 C8.C22 S3×Q8 S3×D4 Q8⋊3D6 D4.D6 kernel C8⋊(C4×S3) C6.Q16 C12.Q8 C24⋊1C4 C3×C4.Q8 S3×C4⋊C4 C4⋊C4⋊7S3 C2×C8⋊S3 C8⋊S3 C4.Q8 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C8 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 2 1 4 1 1 1 1 2 2

Matrix representation of C8⋊(C4×S3) in GL6(𝔽73)

 32 55 0 0 0 0 61 41 0 0 0 0 0 0 0 0 68 63 0 0 0 0 10 5 0 0 39 5 5 10 0 0 68 34 63 68
,
 46 2 0 0 0 0 0 27 0 0 0 0 0 0 62 0 52 0 0 0 0 62 0 52 0 0 37 0 11 0 0 0 0 37 0 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72

G:=sub<GL(6,GF(73))| [32,61,0,0,0,0,55,41,0,0,0,0,0,0,0,0,39,68,0,0,0,0,5,34,0,0,68,10,5,63,0,0,63,5,10,68],[46,0,0,0,0,0,2,27,0,0,0,0,0,0,62,0,37,0,0,0,0,62,0,37,0,0,52,0,11,0,0,0,0,52,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C8⋊(C4×S3) in GAP, Magma, Sage, TeX

C_8\rtimes (C_4\times S_3)
% in TeX

G:=Group("C8:(C4xS3)");
// GroupNames label

G:=SmallGroup(192,420);
// by ID

G=gap.SmallGroup(192,420);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,555,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^4=c^3=d^2=1,b*a*b^-1=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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