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## G = C6×C4.10D4order 192 = 26·3

### Direct product of C6 and C4.10D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4.10D4 — C6×C4.10D4
 Lower central C1 — C2 — C22 — C6×C4.10D4
 Upper central C1 — C2×C6 — C22×C12 — C6×C4.10D4

Generators and relations for C6×C4.10D4
G = < a,b,c,d | a6=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 210 in 146 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C4.10D4, C2×M4(2), C22×Q8, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C6×Q8, C6×Q8, C2×C4.10D4, C3×C4.10D4, C6×M4(2), Q8×C2×C6, C6×C4.10D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C4.10D4, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, C2×C4.10D4, C3×C4.10D4, C6×C22⋊C4, C6×C4.10D4

Smallest permutation representation of C6×C4.10D4
On 96 points
Generators in S96
(1 14 87 63 35 21)(2 15 88 64 36 22)(3 16 81 57 37 23)(4 9 82 58 38 24)(5 10 83 59 39 17)(6 11 84 60 40 18)(7 12 85 61 33 19)(8 13 86 62 34 20)(25 46 74 50 66 90)(26 47 75 51 67 91)(27 48 76 52 68 92)(28 41 77 53 69 93)(29 42 78 54 70 94)(30 43 79 55 71 95)(31 44 80 56 72 96)(32 45 73 49 65 89)
(1 61 5 57)(2 58 6 62)(3 63 7 59)(4 60 8 64)(9 40 13 36)(10 37 14 33)(11 34 15 38)(12 39 16 35)(17 81 21 85)(18 86 22 82)(19 83 23 87)(20 88 24 84)(25 52 29 56)(26 49 30 53)(27 54 31 50)(28 51 32 55)(41 67 45 71)(42 72 46 68)(43 69 47 65)(44 66 48 70)(73 95 77 91)(74 92 78 96)(75 89 79 93)(76 94 80 90)
(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 48 57 66 5 44 61 70)(2 65 62 47 6 69 58 43)(3 46 59 72 7 42 63 68)(4 71 64 45 8 67 60 41)(9 95 36 73 13 91 40 77)(10 80 33 94 14 76 37 90)(11 93 38 79 15 89 34 75)(12 78 35 92 16 74 39 96)(17 31 85 54 21 27 81 50)(18 53 82 30 22 49 86 26)(19 29 87 52 23 25 83 56)(20 51 84 28 24 55 88 32)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 C3×D4 C4.10D4 C3×C4.10D4 kernel C6×C4.10D4 C3×C4.10D4 C6×M4(2) Q8×C2×C6 C2×C4.10D4 C22×C12 C6×Q8 C4.10D4 C2×M4(2) C22×Q8 C22×C4 C2×Q8 C2×C12 C2×C4 C6 C2 # reps 1 4 2 1 2 4 4 8 4 2 8 8 4 8 2 4

Matrix representation of C6×C4.10D4 in GL6(𝔽73)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 1 72 0 72 0 0 0 72 1 0
,
 28 63 0 0 0 0 42 45 0 0 0 0 0 0 1 0 71 0 0 0 0 0 72 1 0 0 0 1 72 0 0 0 1 0 72 0
,
 7 60 0 0 0 0 43 66 0 0 0 0 0 0 36 0 41 33 0 0 16 0 4 37 0 0 20 16 57 53 0 0 0 20 57 53

G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,71,72,72,72,0,0,0,0,0,1,0,0,0,0,72,0],[28,42,0,0,0,0,63,45,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,71,72,72,72,0,0,0,1,0,0],[7,43,0,0,0,0,60,66,0,0,0,0,0,0,36,16,20,0,0,0,0,0,16,20,0,0,41,4,57,57,0,0,33,37,53,53] >;

C6×C4.10D4 in GAP, Magma, Sage, TeX

C_6\times C_4._{10}D_4
% in TeX

G:=Group("C6xC4.10D4");
// GroupNames label

G:=SmallGroup(192,845);
// by ID

G=gap.SmallGroup(192,845);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,4204,3036,124]);
// Polycyclic

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

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