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## G = C2×C6.D8order 192 = 26·3

### Direct product of C2 and C6.D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C6.D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C22×D12 — C2×C6.D8
 Lower central C3 — C6 — C12 — C2×C6.D8
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×C6.D8
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 696 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×10], C23, C23 [×10], C12 [×2], C12 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×6], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C22×C4, C22×C4, C2×D4 [×9], C24, C3⋊C8 [×2], D12 [×4], D12 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×S3 [×10], C22×C6, D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4, C2×D12 [×6], C2×D12 [×3], C22×C12, C22×C12, S3×C23, C2×D4⋊C4, C6.D8 [×4], C22×C3⋊C8, C6×C4⋊C4, C22×D12, C2×C6.D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, D6⋊C4 [×4], D4⋊S3 [×2], Q82S3 [×2], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×D4⋊C4, C6.D8 [×4], C2×D6⋊C4, C2×D4⋊S3, C2×Q82S3, C2×C6.D8

Smallest permutation representation of C2×C6.D8
On 96 points
Generators in S96
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 93)(42 94)(43 95)(44 96)(45 89)(46 90)(47 91)(48 92)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(1 53 47 15 81 75)(2 76 82 16 48 54)(3 55 41 9 83 77)(4 78 84 10 42 56)(5 49 43 11 85 79)(6 80 86 12 44 50)(7 51 45 13 87 73)(8 74 88 14 46 52)(17 37 27 72 90 57)(18 58 91 65 28 38)(19 39 29 66 92 59)(20 60 93 67 30 40)(21 33 31 68 94 61)(22 62 95 69 32 34)(23 35 25 70 96 63)(24 64 89 71 26 36)
(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 22)(2 68)(3 20)(4 66)(5 18)(6 72)(7 24)(8 70)(9 67)(10 19)(11 65)(12 17)(13 71)(14 23)(15 69)(16 21)(25 74)(26 45)(27 80)(28 43)(29 78)(30 41)(31 76)(32 47)(33 82)(34 53)(35 88)(36 51)(37 86)(38 49)(39 84)(40 55)(42 59)(44 57)(46 63)(48 61)(50 90)(52 96)(54 94)(56 92)(58 79)(60 77)(62 75)(64 73)(81 95)(83 93)(85 91)(87 89)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 D8 SD16 C4×S3 D12 C3⋊D4 C3⋊D4 D4⋊S3 Q8⋊2S3 kernel C2×C6.D8 C6.D8 C22×C3⋊C8 C6×C4⋊C4 C22×D12 C2×D12 C2×C4⋊C4 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 3 1 2 1 4 4 4 4 2 2 2 2

Matrix representation of C2×C6.D8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 72 0 0 0 0 1 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 46 0 0 0 0 46 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 16 16 0 0 0 0 57 16
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,16,57,0,0,0,0,16,16],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C6.D8 in GAP, Magma, Sage, TeX

C_2\times C_6.D_8
% in TeX

G:=Group("C2xC6.D8");
// GroupNames label

G:=SmallGroup(192,524);
// by ID

G=gap.SmallGroup(192,524);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1684,438,102,6278]);
// Polycyclic

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

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