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## G = C2×C12.53D4order 192 = 26·3

### Direct product of C2 and C12.53D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C12.53D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C2×C12.53D4
 Lower central C3 — C6 — C12 — C2×C12.53D4
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C2×C12.53D4
G = < a,b,c,d | a2=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c3 >

Subgroups: 184 in 106 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C23, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C3⋊C8, C24, C2×C12, C22×C6, C8.C4, C22×C8, C2×M4(2), C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C2×C8.C4, C12.53D4, C22×C3⋊C8, C2×C4.Dic3, C6×M4(2), C2×C12.53D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C8.C4, C2×C4⋊C4, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C8.C4, C12.53D4, C2×Dic3⋊C4, C2×C12.53D4

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 ··· 8 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 6 ··· 6 12 12 12 12 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + - - + + - - image C1 C2 C2 C2 C2 C4 S3 D4 Q8 Q8 D6 D6 Dic6 C4×S3 C3⋊D4 Dic6 C8.C4 C12.53D4 kernel C2×C12.53D4 C12.53D4 C22×C3⋊C8 C2×C4.Dic3 C6×M4(2) C2×C3⋊C8 C2×M4(2) C2×C12 C2×C12 C22×C6 M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 8 1 2 1 1 2 1 2 4 4 2 8 4

Matrix representation of C2×C12.53D4 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 72 0 0 0 1 72 0 0 0 0 0 46 0 0 0 0 0 46
,
 1 0 0 0 0 0 7 59 0 0 0 66 66 0 0 0 0 0 22 0 0 0 0 0 10
,
 72 0 0 0 0 0 66 14 0 0 0 7 7 0 0 0 0 0 0 51 0 0 0 51 0

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,72,72,0,0,0,0,0,46,0,0,0,0,0,46],[1,0,0,0,0,0,7,66,0,0,0,59,66,0,0,0,0,0,22,0,0,0,0,0,10],[72,0,0,0,0,0,66,7,0,0,0,14,7,0,0,0,0,0,0,51,0,0,0,51,0] >;

C2×C12.53D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{53}D_4
% in TeX

G:=Group("C2xC12.53D4");
// GroupNames label

G:=SmallGroup(192,682);
// by ID

G=gap.SmallGroup(192,682);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,58,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^3>;
// generators/relations

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