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## G = (C2×C8).200D6order 192 = 26·3

### 9th non-split extension by C2×C8 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C8).200D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C23.12D6 — (C2×C8).200D6
 Lower central C3 — C6 — C2×C12 — (C2×C8).200D6
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for (C2×C8).200D6
G = < a,b,c,d | a2=b8=c6=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab-1, dcd-1=ab4c-1 >

Subgroups: 264 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C4×C8, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C6×D4, C42.78C22, C6.SD16, C8×Dic3, C2.Dic12, D4⋊Dic3, C3×D4⋊C4, C4.Dic6, C23.12D6, (C2×C8).200D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C4○D8, C4○D12, S3×D4, D42S3, C42.78C22, C23.11D6, D83S3, Q8.7D6, (C2×C8).200D6

Smallest permutation representation of (C2×C8).200D6
On 96 points
Generators in S96
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 73)
(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 51 42)(2 14 43 88 52 74)(3 49 44 7 53 48)(4 12 45 86 54 80)(5 55 46)(6 10 47 84 56 78)(8 16 41 82 50 76)(9 73 83 13 77 87)(11 79 85)(15 75 81)(17 34 89 23 36 95)(18 58 90 70 37 32)(19 40 91 21 38 93)(20 64 92 68 39 30)(22 62 94 66 33 28)(24 60 96 72 35 26)(25 65 59 27 71 61)(29 69 63 31 67 57)
(1 22 5 18)(2 69 6 65)(3 20 7 24)(4 67 8 71)(9 30 13 26)(10 91 14 95)(11 28 15 32)(12 89 16 93)(17 82 21 86)(19 88 23 84)(25 54 29 50)(27 52 31 56)(33 46 37 42)(34 78 38 74)(35 44 39 48)(36 76 40 80)(41 59 45 63)(43 57 47 61)(49 96 53 92)(51 94 55 90)(58 79 62 75)(60 77 64 73)(66 81 70 85)(68 87 72 83)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 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 C2 C2 S3 D4 D6 D6 D6 C4○D4 C4○D8 C4○D12 D4⋊2S3 S3×D4 D8⋊3S3 Q8.7D6 kernel (C2×C8).200D6 C6.SD16 C8×Dic3 C2.Dic12 D4⋊Dic3 C3×D4⋊C4 C4.Dic6 C23.12D6 D4⋊C4 C2×Dic3 C4⋊C4 C2×C8 C2×D4 C12 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 4 8 4 1 1 2 2

Matrix representation of (C2×C8).200D6 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 67 6 0 0 67 67 0 0 0 0 27 0 0 0 0 27
,
 1 0 0 0 0 72 0 0 0 0 8 0 0 0 50 9
,
 46 0 0 0 0 27 0 0 0 0 49 55 0 0 36 24
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[67,67,0,0,6,67,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,72,0,0,0,0,8,50,0,0,0,9],[46,0,0,0,0,27,0,0,0,0,49,36,0,0,55,24] >;

(C2×C8).200D6 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{200}D_6
% in TeX

G:=Group("(C2xC8).200D6");
// GroupNames label

G:=SmallGroup(192,327);
// by ID

G=gap.SmallGroup(192,327);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,1094,135,100,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^6=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^-1,d*c*d^-1=a*b^4*c^-1>;
// generators/relations

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