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## G = C2×C4.D12order 192 = 26·3

### Direct product of C2 and C4.D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C4.D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×C4.D12
 Lower central C3 — C2×C6 — C2×C4.D12
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×C4.D12
G = < a,b,c,d | a2=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 792 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, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C2×C22⋊Q8, C4.D12, C2×C4⋊Dic3, C2×D6⋊C4, C6×C4⋊C4, C22×Dic6, S3×C22×C4, C2×C4.D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C24, D12, C22×S3, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×D12, D42S3, S3×Q8, S3×C23, C2×C22⋊Q8, C4.D12, C22×D12, C2×D42S3, C2×S3×Q8, C2×C4.D12

Smallest permutation representation of C2×C4.D12
On 96 points
Generators in S96
(1 69)(2 70)(3 71)(4 72)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 73)(35 74)(36 75)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 50 17 32)(2 33 18 51)(3 52 19 34)(4 35 20 53)(5 54 21 36)(6 25 22 55)(7 56 23 26)(8 27 24 57)(9 58 13 28)(10 29 14 59)(11 60 15 30)(12 31 16 49)(37 68 82 93)(38 94 83 69)(39 70 84 95)(40 96 73 71)(41 72 74 85)(42 86 75 61)(43 62 76 87)(44 88 77 63)(45 64 78 89)(46 90 79 65)(47 66 80 91)(48 92 81 67)
(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 68 17 93)(2 92 18 67)(3 66 19 91)(4 90 20 65)(5 64 21 89)(6 88 22 63)(7 62 23 87)(8 86 24 61)(9 72 13 85)(10 96 14 71)(11 70 15 95)(12 94 16 69)(25 44 55 77)(26 76 56 43)(27 42 57 75)(28 74 58 41)(29 40 59 73)(30 84 60 39)(31 38 49 83)(32 82 50 37)(33 48 51 81)(34 80 52 47)(35 46 53 79)(36 78 54 45)

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

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

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

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 D12 D4⋊2S3 S3×Q8 kernel C2×C4.D12 C4.D12 C2×C4⋊Dic3 C2×D6⋊C4 C6×C4⋊C4 C22×Dic6 S3×C22×C4 C2×C4⋊C4 C2×C12 C22×S3 C4⋊C4 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 8 2 2 1 1 1 1 4 4 4 3 4 8 2 2

Matrix representation of C2×C4.D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 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
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 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 7 10 0 0 0 0 3 10
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 7 10 0 0 0 0 3 6

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,7,3,0,0,0,0,10,10],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;

C2×C4.D12 in GAP, Magma, Sage, TeX

C_2\times C_4.D_{12}
% in TeX

G:=Group("C2xC4.D12");
// GroupNames label

G:=SmallGroup(192,1068);
// by ID

G=gap.SmallGroup(192,1068);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,192,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

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