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## G = C12×C4○D4order 192 = 26·3

### Direct product of C12 and C4○D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C12×C4○D4
G = < a,b,c,d | a12=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 370 in 310 conjugacy classes, 250 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C4×C4○D4, C2×C4×C12, C3×C42⋊C2, D4×C12, Q8×C12, C6×C4○D4, C12×C4○D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, C24, C2×C12, C22×C6, C23×C4, C2×C4○D4, C22×C12, C3×C4○D4, C23×C6, C4×C4○D4, C23×C12, C6×C4○D4, C12×C4○D4

Smallest permutation representation of C12×C4○D4
On 96 points
Generators in S96
(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 41 49 84)(2 42 50 73)(3 43 51 74)(4 44 52 75)(5 45 53 76)(6 46 54 77)(7 47 55 78)(8 48 56 79)(9 37 57 80)(10 38 58 81)(11 39 59 82)(12 40 60 83)(13 29 70 90)(14 30 71 91)(15 31 72 92)(16 32 61 93)(17 33 62 94)(18 34 63 95)(19 35 64 96)(20 36 65 85)(21 25 66 86)(22 26 67 87)(23 27 68 88)(24 28 69 89)
(1 20 49 65)(2 21 50 66)(3 22 51 67)(4 23 52 68)(5 24 53 69)(6 13 54 70)(7 14 55 71)(8 15 56 72)(9 16 57 61)(10 17 58 62)(11 18 59 63)(12 19 60 64)(25 73 86 42)(26 74 87 43)(27 75 88 44)(28 76 89 45)(29 77 90 46)(30 78 91 47)(31 79 92 48)(32 80 93 37)(33 81 94 38)(34 82 95 39)(35 83 96 40)(36 84 85 41)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(25 92)(26 93)(27 94)(28 95)(29 96)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A ··· 4L 4M ··· 4AD 6A ··· 6F 6G ··· 6R 12A ··· 12X 12Y ··· 12BH order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 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 2 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 C4○D4 C3×C4○D4 kernel C12×C4○D4 C2×C4×C12 C3×C42⋊C2 D4×C12 Q8×C12 C6×C4○D4 C4×C4○D4 C3×C4○D4 C2×C42 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C12 C4 # reps 1 3 3 6 2 1 2 16 6 6 12 4 2 32 8 16

Matrix representation of C12×C4○D4 in GL3(𝔽13) generated by

 8 0 0 0 11 0 0 0 11
,
 12 0 0 0 5 0 0 0 5
,
 1 0 0 0 10 5 0 11 3
,
 1 0 0 0 12 3 0 0 1
G:=sub<GL(3,GF(13))| [8,0,0,0,11,0,0,0,11],[12,0,0,0,5,0,0,0,5],[1,0,0,0,10,11,0,5,3],[1,0,0,0,12,0,0,3,1] >;

C12×C4○D4 in GAP, Magma, Sage, TeX

C_{12}\times C_4\circ D_4
% in TeX

G:=Group("C12xC4oD4");
// GroupNames label

G:=SmallGroup(192,1406);
// by ID

G=gap.SmallGroup(192,1406);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,520,192]);
// Polycyclic

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

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