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## G = C2×D4.D6order 192 = 26·3

### Direct product of C2 and D4.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D4.D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×Q8 — C2×D4.D6
 Lower central C3 — C6 — C12 — C2×D4.D6
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×D4.D6
G = < a,b,c,d,e | a2=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >

Subgroups: 632 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), SD16, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C2×M4(2), C2×SD16, C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C8⋊S3, Dic12, C2×C3⋊C8, D4.S3, C3⋊Q16, C2×C24, C3×SD16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, D42S3, D42S3, S3×Q8, S3×Q8, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C2×C8.C22, C2×C8⋊S3, C2×Dic12, D4.D6, C2×D4.S3, C2×C3⋊Q16, C6×SD16, C2×D42S3, C2×S3×Q8, C2×D4.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C8.C22, C22×D4, S3×D4, S3×C23, C2×C8.C22, D4.D6, C2×S3×D4, C2×D4.D6

Smallest permutation representation of C2×D4.D6
On 96 points
Generators in S96
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 13)(9 14)(10 15)(11 16)(12 17)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 85)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 61)(48 62)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)
(1 76 7 82)(2 83 8 77)(3 78 9 84)(4 73 10 79)(5 80 11 74)(6 75 12 81)(13 49 19 55)(14 56 20 50)(15 51 21 57)(16 58 22 52)(17 53 23 59)(18 60 24 54)(25 63 31 69)(26 70 32 64)(27 65 33 71)(28 72 34 66)(29 67 35 61)(30 62 36 68)(37 92 43 86)(38 87 44 93)(39 94 45 88)(40 89 46 95)(41 96 47 90)(42 91 48 85)
(1 42)(2 86)(3 44)(4 88)(5 46)(6 90)(7 48)(8 92)(9 38)(10 94)(11 40)(12 96)(13 31)(14 64)(15 33)(16 66)(17 35)(18 68)(19 25)(20 70)(21 27)(22 72)(23 29)(24 62)(26 50)(28 52)(30 54)(32 56)(34 58)(36 60)(37 77)(39 79)(41 81)(43 83)(45 73)(47 75)(49 63)(51 65)(53 67)(55 69)(57 71)(59 61)(74 95)(76 85)(78 87)(80 89)(82 91)(84 93)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 22 19 16)(14 15 20 21)(17 18 23 24)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 58 55 52)(50 51 56 57)(53 54 59 60)(61 68 67 62)(63 66 69 72)(64 71 70 65)(73 78 79 84)(74 83 80 77)(75 76 81 82)(85 96 91 90)(86 89 92 95)(87 94 93 88)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 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 C2 S3 D4 D4 D4 D6 D6 D6 D6 C8.C22 S3×D4 S3×D4 D4.D6 kernel C2×D4.D6 C2×C8⋊S3 C2×Dic12 D4.D6 C2×D4.S3 C2×C3⋊Q16 C6×SD16 C2×D4⋊2S3 C2×S3×Q8 C2×SD16 C4×S3 C2×Dic3 C22×S3 C2×C8 SD16 C2×D4 C2×Q8 C6 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 1 2 1 1 1 4 1 1 2 1 1 4

Matrix representation of C2×D4.D6 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 72 0 0 0 0 0 0 72 0 0
,
 1 72 0 0 0 0 1 0 0 0 0 0 0 0 56 41 0 0 0 0 41 17 0 0 0 0 0 0 41 17 0 0 0 0 17 32
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 56 41 0 0 0 0 41 17 0 0 0 0 0 0 32 56 0 0 0 0 56 41

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,0,0,0,0,0,0,72,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,56,41,0,0,0,0,41,17,0,0,0,0,0,0,41,17,0,0,0,0,17,32],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,56,41,0,0,0,0,41,17,0,0,0,0,0,0,32,56,0,0,0,0,56,41] >;

C2×D4.D6 in GAP, Magma, Sage, TeX

C_2\times D_4.D_6
% in TeX

G:=Group("C2xD4.D6");
// GroupNames label

G:=SmallGroup(192,1319);
// by ID

G=gap.SmallGroup(192,1319);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,1123,185,438,235,102,6278]);
// Polycyclic

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

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