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## G = C2×D6.C8order 192 = 26·3

### Direct product of C2 and D6.C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D6.C8
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×C2×C8 — C2×D6.C8
 Lower central C3 — C6 — C2×D6.C8
 Upper central C1 — C2×C8 — C2×C16

Generators and relations for C2×D6.C8
G = < a,b,c,d | a2=b6=c2=1, d8=b3, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 168 in 90 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C16, C16, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C16, C2×C16, M5(2), C22×C8, C3⋊C16, C48, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C2×M5(2), D6.C8, C2×C3⋊C16, C2×C48, S3×C2×C8, C2×D6.C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, M5(2), C22×C8, S3×C8, S3×C2×C4, C2×M5(2), D6.C8, S3×C2×C8, C2×D6.C8

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A ··· 8H 8I 8J 8K 8L 12A 12B 12C 12D 16A ··· 16H 16I ··· 16P 24A ··· 24H 48A ··· 48P order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 8 ··· 8 8 8 8 8 12 12 12 12 16 ··· 16 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 6 6 2 1 1 1 1 6 6 2 2 2 1 ··· 1 6 6 6 6 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 S3 D6 D6 C4×S3 C4×S3 M5(2) S3×C8 S3×C8 D6.C8 kernel C2×D6.C8 D6.C8 C2×C3⋊C16 C2×C48 S3×C2×C8 S3×C8 C2×C3⋊C8 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 C2×C16 C16 C2×C8 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 8 4 4 1 2 1 2 2 8 4 4 16

Matrix representation of C2×D6.C8 in GL4(𝔽97) generated by

 1 0 0 0 0 1 0 0 0 0 96 0 0 0 0 96
,
 96 0 0 0 0 96 0 0 0 0 1 96 0 0 1 0
,
 96 0 0 0 0 1 0 0 0 0 1 0 0 0 1 96
,
 0 1 0 0 33 0 0 0 0 0 40 17 0 0 80 57
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,96,0,0,0,0,96],[96,0,0,0,0,96,0,0,0,0,1,1,0,0,96,0],[96,0,0,0,0,1,0,0,0,0,1,1,0,0,0,96],[0,33,0,0,1,0,0,0,0,0,40,80,0,0,17,57] >;

C2×D6.C8 in GAP, Magma, Sage, TeX

C_2\times D_6.C_8
% in TeX

G:=Group("C2xD6.C8");
// GroupNames label

G:=SmallGroup(192,459);
// by ID

G=gap.SmallGroup(192,459);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,58,80,102,6278]);
// Polycyclic

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

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