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## G = C6×C4⋊1D4order 192 = 26·3

### Direct product of C6 and C4⋊1D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C4⋊1D4
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C4⋊1D4 — C6×C4⋊1D4
 Lower central C1 — C22 — C6×C4⋊1D4
 Upper central C1 — C22×C6 — C6×C4⋊1D4

Generators and relations for C6×C41D4
G = < a,b,c,d | a6=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 882 in 498 conjugacy classes, 210 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C42, C22×C4, C2×D4, C2×D4, C24, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C42, C41D4, C22×D4, C4×C12, C22×C12, C6×D4, C6×D4, C23×C6, C2×C41D4, C2×C4×C12, C3×C41D4, D4×C2×C6, C6×C41D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C41D4, C22×D4, C6×D4, C23×C6, C2×C41D4, C3×C41D4, D4×C2×C6, C6×C41D4

Smallest permutation representation of C6×C41D4
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 8 31 92)(2 9 32 93)(3 10 33 94)(4 11 34 95)(5 12 35 96)(6 7 36 91)(13 38 20 25)(14 39 21 26)(15 40 22 27)(16 41 23 28)(17 42 24 29)(18 37 19 30)(43 86 62 67)(44 87 63 68)(45 88 64 69)(46 89 65 70)(47 90 66 71)(48 85 61 72)(49 84 60 73)(50 79 55 74)(51 80 56 75)(52 81 57 76)(53 82 58 77)(54 83 59 78)
(1 66 26 58)(2 61 27 59)(3 62 28 60)(4 63 29 55)(5 64 30 56)(6 65 25 57)(7 70 13 76)(8 71 14 77)(9 72 15 78)(10 67 16 73)(11 68 17 74)(12 69 18 75)(19 80 96 88)(20 81 91 89)(21 82 92 90)(22 83 93 85)(23 84 94 86)(24 79 95 87)(31 47 39 53)(32 48 40 54)(33 43 41 49)(34 44 42 50)(35 45 37 51)(36 46 38 52)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 67)(7 62)(8 63)(9 64)(10 65)(11 66)(12 61)(13 60)(14 55)(15 56)(16 57)(17 58)(18 59)(19 54)(20 49)(21 50)(22 51)(23 52)(24 53)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 87)(32 88)(33 89)(34 90)(35 85)(36 86)(37 83)(38 84)(39 79)(40 80)(41 81)(42 82)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A ··· 4L 6A ··· 6N 6O ··· 6AD 12A ··· 12X order 1 2 ··· 2 2 ··· 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 4 ··· 4 1 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel C6×C4⋊1D4 C2×C4×C12 C3×C4⋊1D4 D4×C2×C6 C2×C4⋊1D4 C2×C42 C4⋊1D4 C22×D4 C2×C12 C2×C4 # reps 1 1 8 6 2 2 16 12 12 24

Matrix representation of C6×C41D4 in GL6(𝔽13)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1

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

C6×C41D4 in GAP, Magma, Sage, TeX

C_6\times C_4\rtimes_1D_4
% in TeX

G:=Group("C6xC4:1D4");
// GroupNames label

G:=SmallGroup(192,1419);
// by ID

G=gap.SmallGroup(192,1419);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102,520]);
// Polycyclic

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

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