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## G = C8.27(C4×S3)  order 192 = 26·3

### 4th non-split extension by C8 of C4×S3 acting via C4×S3/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C8.27(C4×S3)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — S3×C2×C8 — C8.27(C4×S3)
 Lower central C3 — C6 — C12 — C8.27(C4×S3)
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C8.27(C4×S3)
G = < a,b,c,d | a8=b4=c3=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 272 in 114 conjugacy classes, 55 normal (27 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, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.Q8, C2.D8, C2.D8, C42⋊C2, C22×C8, S3×C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C23.25D4, C12.Q8, C241C4, C3×C2.D8, C4⋊C47S3, S3×C2×C8, C8.27(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, C4○D8, S3×C2×C4, S3×D4, S3×Q8, C23.25D4, S3×C4⋊C4, D83S3, D24⋊C2, C8.27(C4×S3)

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D4 D4 D6 D6 C4×S3 C4○D8 S3×Q8 S3×D4 D8⋊3S3 D24⋊C2 kernel C8.27(C4×S3) C12.Q8 C24⋊1C4 C3×C2.D8 C4⋊C4⋊7S3 S3×C2×C8 S3×C8 C2.D8 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C8 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 2 1 1 2 1 4 8 1 1 2 2

Matrix representation of C8.27(C4×S3) in GL6(𝔽73)

 22 0 0 0 0 0 0 10 0 0 0 0 0 0 10 16 0 0 0 0 0 22 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 46 72 0 0 0 0 71 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 46 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [22,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,16,22,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,46,71,0,0,0,0,72,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,46,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C8.27(C4×S3) in GAP, Magma, Sage, TeX

C_8._{27}(C_4\times S_3)
% in TeX

G:=Group("C8.27(C4xS3)");
// GroupNames label

G:=SmallGroup(192,439);
// by ID

G=gap.SmallGroup(192,439);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,58,438,102,6278]);
// Polycyclic

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

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