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G = C6×C4○D8order 192 = 26·3

Direct product of C6 and C4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×C4○D8
 Chief series C1 — C2 — C4 — C12 — C3×D4 — C3×D8 — C3×C4○D8 — C6×C4○D8
 Lower central C1 — C2 — C4 — C6×C4○D8
 Upper central C1 — C2×C12 — C22×C12 — C6×C4○D8

Generators and relations for C6×C4○D8
G = < a,b,c,d | a6=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 402 in 266 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C24, C2×C24, C3×D8, C3×SD16, C3×Q16, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C2×C4○D8, C22×C24, C6×D8, C6×SD16, C6×Q16, C3×C4○D8, C6×C4○D4, C6×C4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C4○D8, C22×D4, C6×D4, C23×C6, C2×C4○D8, C3×C4○D8, D4×C2×C6, C6×C4○D8

Smallest permutation representation of C6×C4○D8
On 96 points
Generators in S96
(1 19 57 39 77 81)(2 20 58 40 78 82)(3 21 59 33 79 83)(4 22 60 34 80 84)(5 23 61 35 73 85)(6 24 62 36 74 86)(7 17 63 37 75 87)(8 18 64 38 76 88)(9 29 95 41 56 71)(10 30 96 42 49 72)(11 31 89 43 50 65)(12 32 90 44 51 66)(13 25 91 45 52 67)(14 26 92 46 53 68)(15 27 93 47 54 69)(16 28 94 48 55 70)
(1 89 5 93)(2 90 6 94)(3 91 7 95)(4 92 8 96)(9 79 13 75)(10 80 14 76)(11 73 15 77)(12 74 16 78)(17 41 21 45)(18 42 22 46)(19 43 23 47)(20 44 24 48)(25 87 29 83)(26 88 30 84)(27 81 31 85)(28 82 32 86)(33 67 37 71)(34 68 38 72)(35 69 39 65)(36 70 40 66)(49 60 53 64)(50 61 54 57)(51 62 55 58)(52 63 56 59)
(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 35)(2 34)(3 33)(4 40)(5 39)(6 38)(7 37)(8 36)(9 41)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 75)(18 74)(19 73)(20 80)(21 79)(22 78)(23 77)(24 76)(25 52)(26 51)(27 50)(28 49)(29 56)(30 55)(31 54)(32 53)(57 85)(58 84)(59 83)(60 82)(61 81)(62 88)(63 87)(64 86)(65 93)(66 92)(67 91)(68 90)(69 89)(70 96)(71 95)(72 94)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C4○D8 C3×C4○D8 kernel C6×C4○D8 C22×C24 C6×D8 C6×SD16 C6×Q16 C3×C4○D8 C6×C4○D4 C2×C4○D8 C22×C8 C2×D8 C2×SD16 C2×Q16 C4○D8 C2×C4○D4 C2×C12 C22×C6 C2×C4 C23 C6 C2 # reps 1 1 1 2 1 8 2 2 2 2 4 2 16 4 3 1 6 2 8 16

Matrix representation of C6×C4○D8 in GL3(𝔽73) generated by

 65 0 0 0 1 0 0 0 1
,
 72 0 0 0 46 0 0 0 46
,
 72 0 0 0 16 57 0 16 16
,
 1 0 0 0 72 0 0 0 1
G:=sub<GL(3,GF(73))| [65,0,0,0,1,0,0,0,1],[72,0,0,0,46,0,0,0,46],[72,0,0,0,16,16,0,57,16],[1,0,0,0,72,0,0,0,1] >;

C6×C4○D8 in GAP, Magma, Sage, TeX

C_6\times C_4\circ D_8
% in TeX

G:=Group("C6xC4oD8");
// GroupNames label

G:=SmallGroup(192,1461);
// by ID

G=gap.SmallGroup(192,1461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,520,6053,3036,124]);
// Polycyclic

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

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