Copied to
clipboard

## G = C6×C8○D4order 192 = 26·3

### Direct product of C6 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×C8○D4
 Chief series C1 — C2 — C4 — C12 — C24 — C2×C24 — C3×C8○D4 — C6×C8○D4
 Lower central C1 — C2 — C6×C8○D4
 Upper central C1 — C2×C24 — C6×C8○D4

Generators and relations for C6×C8○D4
G = < a,b,c,d | a6=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 290 in 266 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×C24, C2×C24, C3×M4(2), C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2×C8○D4, C22×C24, C6×M4(2), C3×C8○D4, C6×C4○D4, C6×C8○D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C24, C2×C12, C22×C6, C8○D4, C23×C4, C22×C12, C23×C6, C2×C8○D4, C3×C8○D4, C23×C12, C6×C8○D4

Smallest permutation representation of C6×C8○D4
On 96 points
Generators in S96
(1 75 87 22 55 47)(2 76 88 23 56 48)(3 77 81 24 49 41)(4 78 82 17 50 42)(5 79 83 18 51 43)(6 80 84 19 52 44)(7 73 85 20 53 45)(8 74 86 21 54 46)(9 30 94 38 62 65)(10 31 95 39 63 66)(11 32 96 40 64 67)(12 25 89 33 57 68)(13 26 90 34 58 69)(14 27 91 35 59 70)(15 28 92 36 60 71)(16 29 93 37 61 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 91 5 95)(2 92 6 96)(3 93 7 89)(4 94 8 90)(9 54 13 50)(10 55 14 51)(11 56 15 52)(12 49 16 53)(17 65 21 69)(18 66 22 70)(19 67 23 71)(20 68 24 72)(25 41 29 45)(26 42 30 46)(27 43 31 47)(28 44 32 48)(33 77 37 73)(34 78 38 74)(35 79 39 75)(36 80 40 76)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 78)(10 79)(11 80)(12 73)(13 74)(14 75)(15 76)(16 77)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A 4B 4C 4D 4E ··· 4J 6A ··· 6F 6G ··· 6R 8A ··· 8H 8I ··· 8T 12A ··· 12H 12I ··· 12T 24A ··· 24P 24Q ··· 24AN order 1 2 2 2 2 ··· 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 1 1 2 ··· 2 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 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 type + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C12 C12 C12 C8○D4 C3×C8○D4 kernel C6×C8○D4 C22×C24 C6×M4(2) C3×C8○D4 C6×C4○D4 C2×C8○D4 C6×D4 C6×Q8 C3×C4○D4 C22×C8 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C6 C2 # reps 1 3 3 8 1 2 6 2 8 6 6 16 2 12 4 16 8 16

Matrix representation of C6×C8○D4 in GL3(𝔽73) generated by

 72 0 0 0 8 0 0 0 8
,
 27 0 0 0 22 0 0 0 22
,
 1 0 0 0 46 0 0 2 27
,
 1 0 0 0 46 1 0 2 27
G:=sub<GL(3,GF(73))| [72,0,0,0,8,0,0,0,8],[27,0,0,0,22,0,0,0,22],[1,0,0,0,46,2,0,0,27],[1,0,0,0,46,2,0,1,27] >;

C6×C8○D4 in GAP, Magma, Sage, TeX

C_6\times C_8\circ D_4
% in TeX

G:=Group("C6xC8oD4");
// GroupNames label

G:=SmallGroup(192,1456);
// by ID

G=gap.SmallGroup(192,1456);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,1059,124]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

׿
×
𝔽