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## G = C3×C4○D16order 192 = 26·3

### Direct product of C3 and C4○D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×C4○D16
 Chief series C1 — C2 — C4 — C8 — C24 — C3×D8 — C3×D16 — C3×C4○D16
 Lower central C1 — C2 — C4 — C8 — C3×C4○D16
 Upper central C1 — C12 — C2×C12 — C2×C24 — C3×C4○D16

Generators and relations for C3×C4○D16
G = < a,b,c,d | a3=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >

Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C2×C16, D16, SD32, Q32, C4○D8, C48, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C4○D16, C2×C48, C3×D16, C3×SD32, C3×Q32, C3×C4○D8, C3×C4○D16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, C2×D8, C3×D8, C6×D4, C4○D16, C6×D8, C3×C4○D16

Smallest permutation representation of C3×C4○D16
On 96 points
Generators in S96
(1 24 72)(2 25 73)(3 26 74)(4 27 75)(5 28 76)(6 29 77)(7 30 78)(8 31 79)(9 32 80)(10 17 65)(11 18 66)(12 19 67)(13 20 68)(14 21 69)(15 22 70)(16 23 71)(33 50 85)(34 51 86)(35 52 87)(36 53 88)(37 54 89)(38 55 90)(39 56 91)(40 57 92)(41 58 93)(42 59 94)(43 60 95)(44 61 96)(45 62 81)(46 63 82)(47 64 83)(48 49 84)
(1 33 9 41)(2 34 10 42)(3 35 11 43)(4 36 12 44)(5 37 13 45)(6 38 14 46)(7 39 15 47)(8 40 16 48)(17 59 25 51)(18 60 26 52)(19 61 27 53)(20 62 28 54)(21 63 29 55)(22 64 30 56)(23 49 31 57)(24 50 32 58)(65 94 73 86)(66 95 74 87)(67 96 75 88)(68 81 76 89)(69 82 77 90)(70 83 78 91)(71 84 79 92)(72 85 80 93)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 32)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 50)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 80)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 2 8 8 1 1 1 1 2 8 8 1 1 2 2 8 8 8 8 2 2 2 2 1 1 1 1 2 2 8 8 8 8 2 ··· 2 2 ··· 2 2 ··· 2

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D8 D8 C3×D4 C3×D4 C3×D8 C3×D8 C4○D16 C3×C4○D16 kernel C3×C4○D16 C2×C48 C3×D16 C3×SD32 C3×Q32 C3×C4○D8 C4○D16 C2×C16 D16 SD32 Q32 C4○D8 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 2 2 2 4 4 8 16

Matrix representation of C3×C4○D16 in GL2(𝔽97) generated by

 35 0 0 35
,
 75 0 0 75
,
 2 26 71 2
,
 71 2 2 26
G:=sub<GL(2,GF(97))| [35,0,0,35],[75,0,0,75],[2,71,26,2],[71,2,2,26] >;

C3×C4○D16 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{16}
% in TeX

G:=Group("C3xC4oD16");
// GroupNames label

G:=SmallGroup(192,941);
// by ID

G=gap.SmallGroup(192,941);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,520,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^8=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^7>;
// generators/relations

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