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G = C3×C4.D8order 192 = 26·3

Direct product of C3 and C4.D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C4.D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C12 — C3×C4⋊C8 — C3×C4.D8
 Lower central C1 — C22 — C2×C4 — C3×C4.D8
 Upper central C1 — C2×C6 — C4×C12 — C3×C4.D8

Generators and relations for C3×C4.D8
G = < a,b,c,d | a3=b4=c8=1, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=bc-1 >

Subgroups: 210 in 84 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C4⋊C8, C41D4, C4×C12, C2×C24, C6×D4, C6×D4, C4.D8, C3×C4⋊C8, C3×C41D4, C3×C4.D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, C2×C12, C3×D4, C4.D4, D4⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, C4.D8, C3×C4.D4, C3×D4⋊C4, C3×C4.D8

Smallest permutation representation of C3×C4.D8
On 96 points
Generators in S96
(1 35 25)(2 36 26)(3 37 27)(4 38 28)(5 39 29)(6 40 30)(7 33 31)(8 34 32)(9 89 82)(10 90 83)(11 91 84)(12 92 85)(13 93 86)(14 94 87)(15 95 88)(16 96 81)(17 74 67)(18 75 68)(19 76 69)(20 77 70)(21 78 71)(22 79 72)(23 80 65)(24 73 66)(41 57 53)(42 58 54)(43 59 55)(44 60 56)(45 61 49)(46 62 50)(47 63 51)(48 64 52)
(1 23 82 46)(2 47 83 24)(3 17 84 48)(4 41 85 18)(5 19 86 42)(6 43 87 20)(7 21 88 44)(8 45 81 22)(9 62 35 80)(10 73 36 63)(11 64 37 74)(12 75 38 57)(13 58 39 76)(14 77 40 59)(15 60 33 78)(16 79 34 61)(25 65 89 50)(26 51 90 66)(27 67 91 52)(28 53 92 68)(29 69 93 54)(30 55 94 70)(31 71 95 56)(32 49 96 72)
(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 23 22 82 8 46 45)(2 44 47 7 83 21 24 88)(3 87 17 20 84 6 48 43)(4 42 41 5 85 19 18 86)(9 34 62 61 35 16 80 79)(10 78 73 15 36 60 63 33)(11 40 64 59 37 14 74 77)(12 76 75 13 38 58 57 39)(25 96 65 72 89 32 50 49)(26 56 51 31 90 71 66 95)(27 94 67 70 91 30 52 55)(28 54 53 29 92 69 68 93)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 6A ··· 6F 6G 6H 6I 6J 8A ··· 8H 12A ··· 12H 12I 12J 24A ··· 24P order 1 2 2 2 2 2 3 3 4 4 4 4 4 6 ··· 6 6 6 6 6 8 ··· 8 12 ··· 12 12 12 24 ··· 24 size 1 1 1 1 8 8 1 1 2 2 2 2 4 1 ··· 1 8 8 8 8 4 ··· 4 2 ··· 2 4 4 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C3 C4 C6 C6 C12 D4 D8 SD16 C3×D4 C3×D8 C3×SD16 C4.D4 C3×C4.D4 kernel C3×C4.D8 C3×C4⋊C8 C3×C4⋊1D4 C4.D8 C6×D4 C4⋊C8 C4⋊1D4 C2×D4 C2×C12 C12 C12 C2×C4 C4 C4 C6 C2 # reps 1 2 1 2 4 4 2 8 2 4 4 4 8 8 1 2

Matrix representation of C3×C4.D8 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 72 71 0 0 1 1
,
 16 57 0 0 16 16 0 0 0 0 41 41 0 0 16 32
,
 16 57 0 0 57 57 0 0 0 0 0 41 0 0 16 32
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,1,0,0,71,1],[16,16,0,0,57,16,0,0,0,0,41,16,0,0,41,32],[16,57,0,0,57,57,0,0,0,0,0,16,0,0,41,32] >;

C3×C4.D8 in GAP, Magma, Sage, TeX

C_3\times C_4.D_8
% in TeX

G:=Group("C3xC4.D8");
// GroupNames label

G:=SmallGroup(192,137);
// by ID

G=gap.SmallGroup(192,137);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951,242]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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