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## G = C3×D4○C16order 192 = 26·3

### Direct product of C3 and D4○C16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×D4○C16
 Chief series C1 — C2 — C4 — C8 — C24 — C48 — C2×C48 — C3×D4○C16
 Lower central C1 — C2 — C3×D4○C16
 Upper central C1 — C48 — C3×D4○C16

Generators and relations for C3×D4○C16
G = < a,b,c,d | a3=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 90 in 84 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C12, C12, C2×C6, C16, C16, C2×C8, M4(2), C4○D4, C24, C24, C2×C12, C3×D4, C3×Q8, C2×C16, M5(2), C8○D4, C48, C48, C2×C24, C3×M4(2), C3×C4○D4, D4○C16, C2×C48, C3×M5(2), C3×C8○D4, C3×D4○C16
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C2×C8, C22×C4, C24, C2×C12, C22×C6, C22×C8, C2×C24, C22×C12, D4○C16, C22×C24, C3×D4○C16

Smallest permutation representation of C3×D4○C16
On 96 points
Generators in S96
(1 89 61)(2 90 62)(3 91 63)(4 92 64)(5 93 49)(6 94 50)(7 95 51)(8 96 52)(9 81 53)(10 82 54)(11 83 55)(12 84 56)(13 85 57)(14 86 58)(15 87 59)(16 88 60)(17 65 34)(18 66 35)(19 67 36)(20 68 37)(21 69 38)(22 70 39)(23 71 40)(24 72 41)(25 73 42)(26 74 43)(27 75 44)(28 76 45)(29 77 46)(30 78 47)(31 79 48)(32 80 33)
(1 78 9 70)(2 79 10 71)(3 80 11 72)(4 65 12 73)(5 66 13 74)(6 67 14 75)(7 68 15 76)(8 69 16 77)(17 56 25 64)(18 57 26 49)(19 58 27 50)(20 59 28 51)(21 60 29 52)(22 61 30 53)(23 62 31 54)(24 63 32 55)(33 83 41 91)(34 84 42 92)(35 85 43 93)(36 86 44 94)(37 87 45 95)(38 88 46 96)(39 89 47 81)(40 90 48 82)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 65)(13 66)(14 67)(15 68)(16 69)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 81)(48 82)
(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)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 8A 8B 8C 8D 8E ··· 8J 12A 12B 12C 12D 12E ··· 12J 16A ··· 16H 16I ··· 16T 24A ··· 24H 24I ··· 24T 48A ··· 48P 48Q ··· 48AN order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 8 8 8 8 8 ··· 8 12 12 12 12 12 ··· 12 16 ··· 16 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 48 ··· 48 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 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 C3 C4 C4 C6 C6 C6 C8 C8 C12 C12 C24 C24 D4○C16 C3×D4○C16 kernel C3×D4○C16 C2×C48 C3×M5(2) C3×C8○D4 D4○C16 C3×M4(2) C3×C4○D4 C2×C16 M5(2) C8○D4 C3×D4 C3×Q8 M4(2) C4○D4 D4 Q8 C3 C1 # reps 1 3 3 1 2 6 2 6 6 2 12 4 12 4 24 8 8 16

Matrix representation of C3×D4○C16 in GL3(𝔽97) generated by

 35 0 0 0 1 0 0 0 1
,
 1 0 0 0 79 80 0 2 18
,
 96 0 0 0 18 16 0 95 79
,
 96 0 0 0 70 0 0 0 70
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[1,0,0,0,79,2,0,80,18],[96,0,0,0,18,95,0,16,79],[96,0,0,0,70,0,0,0,70] >;

C3×D4○C16 in GAP, Magma, Sage, TeX

C_3\times D_4\circ C_{16}
% in TeX

G:=Group("C3xD4oC16");
// GroupNames label

G:=SmallGroup(192,937);
// by ID

G=gap.SmallGroup(192,937);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,1059,102,124]);
// Polycyclic

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

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