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## G = C2×D6⋊3Q8order 192 = 26·3

### Direct product of C2 and D6⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×D6⋊3Q8
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×D6⋊3Q8
 Lower central C3 — C2×C6 — C2×D6⋊3Q8
 Upper central C1 — C23 — C22×Q8

Generators and relations for C2×D63Q8
G = < a,b,c,d,e | a2=b6=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 744 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C6×Q8, C6×Q8, S3×C23, C2×C22⋊Q8, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, D63Q8, S3×C22×C4, Q8×C2×C6, C2×D63Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C24, C3⋊D4, C22×S3, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, S3×Q8, Q83S3, C2×C3⋊D4, S3×C23, C2×C22⋊Q8, D63Q8, C2×S3×Q8, C2×Q83S3, C22×C3⋊D4, C2×D63Q8

Smallest permutation representation of C2×D63Q8
On 96 points
Generators in S96
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 47)(8 48)(9 43)(10 44)(11 45)(12 46)(13 57)(14 58)(15 59)(16 60)(17 55)(18 56)(19 77)(20 78)(21 73)(22 74)(23 75)(24 76)(25 69)(26 70)(27 71)(28 72)(29 67)(30 68)(31 89)(32 90)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(41 79)(42 80)(49 93)(50 94)(51 95)(52 96)(53 91)(54 92)
(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 63)(2 62)(3 61)(4 66)(5 65)(6 64)(7 45)(8 44)(9 43)(10 48)(11 47)(12 46)(13 56)(14 55)(15 60)(16 59)(17 58)(18 57)(19 77)(20 76)(21 75)(22 74)(23 73)(24 78)(25 71)(26 70)(27 69)(28 68)(29 67)(30 72)(31 86)(32 85)(33 90)(34 89)(35 88)(36 87)(37 80)(38 79)(39 84)(40 83)(41 82)(42 81)(49 95)(50 94)(51 93)(52 92)(53 91)(54 96)
(1 28 16 24)(2 29 17 19)(3 30 18 20)(4 25 13 21)(5 26 14 22)(6 27 15 23)(7 83 95 87)(8 84 96 88)(9 79 91 89)(10 80 92 90)(11 81 93 85)(12 82 94 86)(31 43 41 53)(32 44 42 54)(33 45 37 49)(34 46 38 50)(35 47 39 51)(36 48 40 52)(55 77 65 67)(56 78 66 68)(57 73 61 69)(58 74 62 70)(59 75 63 71)(60 76 64 72)
(1 40 16 36)(2 41 17 31)(3 42 18 32)(4 37 13 33)(5 38 14 34)(6 39 15 35)(7 75 95 71)(8 76 96 72)(9 77 91 67)(10 78 92 68)(11 73 93 69)(12 74 94 70)(19 53 29 43)(20 54 30 44)(21 49 25 45)(22 50 26 46)(23 51 27 47)(24 52 28 48)(55 89 65 79)(56 90 66 80)(57 85 61 81)(58 86 62 82)(59 87 63 83)(60 88 64 84)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 C4○D4 C3⋊D4 S3×Q8 Q8⋊3S3 kernel C2×D6⋊3Q8 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×D6⋊C4 D6⋊3Q8 S3×C22×C4 Q8×C2×C6 C22×Q8 C2×C12 C22×S3 C22×C4 C2×Q8 C2×C6 C2×C4 C22 C22 # reps 1 2 1 2 8 1 1 1 4 4 3 4 4 8 2 2

Matrix representation of C2×D63Q8 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 5

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5] >;

C2×D63Q8 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes_3Q_8
% in TeX

G:=Group("C2xD6:3Q8");
// GroupNames label

G:=SmallGroup(192,1372);
// by ID

G=gap.SmallGroup(192,1372);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,136,6278]);
// Polycyclic

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

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