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## G = C2×C8.6D6order 192 = 26·3

### Direct product of C2 and C8.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C2×C8.6D6
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C2×D24 — C2×C8.6D6
 Lower central C3 — C6 — C12 — C24 — C2×C8.6D6
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C2×C8.6D6
G = < a,b,c,d | a2=b8=1, c6=b4, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 344 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C16, C2×C8, D8, Q16, Q16, C2×D4, C2×Q8, C24, D12, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C16, SD32, C2×D8, C2×Q16, C3⋊C16, D24, D24, C2×C24, C3×Q16, C3×Q16, C2×D12, C6×Q8, C2×SD32, C2×C3⋊C16, C8.6D6, C2×D24, C6×Q16, C2×C8.6D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, SD32, C2×D8, D4⋊S3, C2×C3⋊D4, C2×SD32, C8.6D6, C2×D4⋊S3, C2×C8.6D6

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 16A ··· 16H 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 24 24 24 size 1 1 1 1 24 24 2 2 2 8 8 2 2 2 2 2 2 2 4 4 8 8 8 8 6 ··· 6 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 D8 C3⋊D4 C3⋊D4 SD32 D4⋊S3 D4⋊S3 C8.6D6 kernel C2×C8.6D6 C2×C3⋊C16 C8.6D6 C2×D24 C6×Q16 C2×Q16 C24 C2×C12 C2×C8 Q16 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 1 4 1 1 1 1 1 1 2 2 2 2 2 8 1 1 4

Matrix representation of C2×C8.6D6 in GL5(𝔽97)

 96 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 96 0 0 0 0 0 96 0 0 0 0 0 0 76 0 0 0 37 14
,
 96 0 0 0 0 0 15 56 0 0 0 41 56 0 0 0 0 0 43 67 0 0 0 94 54
,
 1 0 0 0 0 0 82 41 0 0 0 56 15 0 0 0 0 0 63 30 0 0 0 58 43

G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,0,37,0,0,0,76,14],[96,0,0,0,0,0,15,41,0,0,0,56,56,0,0,0,0,0,43,94,0,0,0,67,54],[1,0,0,0,0,0,82,56,0,0,0,41,15,0,0,0,0,0,63,58,0,0,0,30,43] >;

C2×C8.6D6 in GAP, Magma, Sage, TeX

C_2\times C_8._6D_6
% in TeX

G:=Group("C2xC8.6D6");
// GroupNames label

G:=SmallGroup(192,737);
// by ID

G=gap.SmallGroup(192,737);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,184,675,185,192,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^6=b^4,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations

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