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## G = D4×C2×C12order 192 = 26·3

### Direct product of C2×C12 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C12
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — D4×C12 — D4×C2×C12
 Lower central C1 — C2 — D4×C2×C12
 Upper central C1 — C22×C12 — D4×C2×C12

Generators and relations for D4×C2×C12
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C22×C12, C6×D4, C23×C6, C2×C4×D4, C2×C4×C12, C6×C22⋊C4, C6×C4⋊C4, D4×C12, C23×C12, D4×C2×C6, D4×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C24, C2×C12, C3×D4, C22×C6, C4×D4, C23×C4, C22×D4, C2×C4○D4, C22×C12, C6×D4, C3×C4○D4, C23×C6, C2×C4×D4, D4×C12, C23×C12, D4×C2×C6, C6×C4○D4, D4×C2×C12

Smallest permutation representation of D4×C2×C12
On 96 points
Generators in S96
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 61)(12 62)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 73)(24 74)(25 95)(26 96)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
(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 49 19 88)(2 50 20 89)(3 51 21 90)(4 52 22 91)(5 53 23 92)(6 54 24 93)(7 55 13 94)(8 56 14 95)(9 57 15 96)(10 58 16 85)(11 59 17 86)(12 60 18 87)(25 70 47 76)(26 71 48 77)(27 72 37 78)(28 61 38 79)(29 62 39 80)(30 63 40 81)(31 64 41 82)(32 65 42 83)(33 66 43 84)(34 67 44 73)(35 68 45 74)(36 69 46 75)
(1 81)(2 82)(3 83)(4 84)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 69)(14 70)(15 71)(16 72)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 95)(26 96)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A ··· 4H 4I ··· 4X 6A ··· 6N 6O ··· 6AD 12A ··· 12P 12Q ··· 12AV order 1 2 ··· 2 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 2 ··· 2 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C6 C12 D4 C4○D4 C3×D4 C3×C4○D4 kernel D4×C2×C12 C2×C4×C12 C6×C22⋊C4 C6×C4⋊C4 D4×C12 C23×C12 D4×C2×C6 C2×C4×D4 C6×D4 C2×C42 C2×C22⋊C4 C2×C4⋊C4 C4×D4 C23×C4 C22×D4 C2×D4 C2×C12 C2×C6 C2×C4 C22 # reps 1 1 2 1 8 2 1 2 16 2 4 2 16 4 2 32 4 4 8 8

Matrix representation of D4×C2×C12 in GL4(𝔽13) generated by

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

D4×C2×C12 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{12}
% in TeX

G:=Group("D4xC2xC12");
// GroupNames label

G:=SmallGroup(192,1404);
// by ID

G=gap.SmallGroup(192,1404);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,520]);
// Polycyclic

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

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