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## G = C2×C6×SD16order 192 = 26·3

### Direct product of C2×C6 and SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C2×C6×SD16
 Chief series C1 — C2 — C4 — C12 — C3×Q8 — C3×SD16 — C6×SD16 — C2×C6×SD16
 Lower central C1 — C2 — C4 — C2×C6×SD16
 Upper central C1 — C22×C6 — C22×C12 — C2×C6×SD16

Generators and relations for C2×C6×SD16
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 498 in 298 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C22×C8, C2×SD16, C22×D4, C22×Q8, C2×C24, C3×SD16, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C6×Q8, C23×C6, C22×SD16, C22×C24, C6×SD16, D4×C2×C6, Q8×C2×C6, C2×C6×SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C24, C3×D4, C22×C6, C2×SD16, C22×D4, C3×SD16, C6×D4, C23×C6, C22×SD16, C6×SD16, D4×C2×C6, C2×C6×SD16

Smallest permutation representation of C2×C6×SD16
On 96 points
Generators in S96
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(33 86)(34 87)(35 88)(36 81)(37 82)(38 83)(39 84)(40 85)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)(65 93)(66 94)(67 95)(68 96)(69 89)(70 90)(71 91)(72 92)
(1 14 82 41 91 54)(2 15 83 42 92 55)(3 16 84 43 93 56)(4 9 85 44 94 49)(5 10 86 45 95 50)(6 11 87 46 96 51)(7 12 88 47 89 52)(8 13 81 48 90 53)(17 70 78 57 28 36)(18 71 79 58 29 37)(19 72 80 59 30 38)(20 65 73 60 31 39)(21 66 74 61 32 40)(22 67 75 62 25 33)(23 68 76 63 26 34)(24 69 77 64 27 35)
(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 41)(2 44)(3 47)(4 42)(5 45)(6 48)(7 43)(8 46)(9 92)(10 95)(11 90)(12 93)(13 96)(14 91)(15 94)(16 89)(17 63)(18 58)(19 61)(20 64)(21 59)(22 62)(23 57)(24 60)(25 67)(26 70)(27 65)(28 68)(29 71)(30 66)(31 69)(32 72)(33 75)(34 78)(35 73)(36 76)(37 79)(38 74)(39 77)(40 80)(49 83)(50 86)(51 81)(52 84)(53 87)(54 82)(55 85)(56 88)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 SD16 C3×D4 C3×D4 C3×SD16 kernel C2×C6×SD16 C22×C24 C6×SD16 D4×C2×C6 Q8×C2×C6 C22×SD16 C22×C8 C2×SD16 C22×D4 C22×Q8 C2×C12 C22×C6 C2×C6 C2×C4 C23 C22 # reps 1 1 12 1 1 2 2 24 2 2 3 1 8 6 2 16

Matrix representation of C2×C6×SD16 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 64 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 72 0 0 0 0 0 61 0 0 6 61
,
 1 0 0 0 0 72 0 0 0 0 72 0 0 0 72 1
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,64,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,6,0,0,61,61],[1,0,0,0,0,72,0,0,0,0,72,72,0,0,0,1] >;

C2×C6×SD16 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm SD}_{16}
% in TeX

G:=Group("C2xC6xSD16");
// GroupNames label

G:=SmallGroup(192,1459);
// by ID

G=gap.SmallGroup(192,1459);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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