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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

Aliases: C3×D4○C16, D4.2C24, Q8.3C24, M5(2)⋊7C6, C48.29C22, C24.83C23, M4(2).4C12, (C2×C16)⋊9C6, (C2×C48)⋊19C2, C16.8(C2×C6), C4.5(C2×C24), C8○D4.5C6, (C3×D4).4C8, (C3×Q8).4C8, C12.34(C2×C8), C24.68(C2×C4), C8.12(C2×C12), C4○D4.6C12, C2.7(C22×C24), C22.1(C2×C24), C8.16(C22×C6), C6.36(C22×C8), (C3×M5(2))⋊15C2, C4.36(C22×C12), (C3×M4(2)).8C4, (C2×C24).449C22, C12.194(C22×C4), (C2×C6).8(C2×C8), (C3×C4○D4).7C4, (C3×C8○D4).6C2, (C2×C4).51(C2×C12), (C2×C8).103(C2×C6), (C2×C12).272(C2×C4), SmallGroup(192,937)

Series: Derived Chief Lower central Upper central

C1C2 — C3×D4○C16
C1C2C4C8C24C48C2×C48 — C3×D4○C16
C1C2 — C3×D4○C16
C1C48 — 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 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C12, C12 [×3], C2×C6 [×3], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C24, C24 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C2×C16 [×3], M5(2) [×3], C8○D4, C48, C48 [×3], C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4, D4○C16, C2×C48 [×3], C3×M5(2) [×3], C3×C8○D4, C3×D4○C16
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C8 [×4], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C2×C8 [×6], C22×C4, C24 [×4], C2×C12 [×6], C22×C6, C22×C8, C2×C24 [×6], C22×C12, D4○C16, C22×C24, C3×D4○C16

Smallest permutation representation of C3×D4○C16
On 96 points
Generators in S96
(1 63 89)(2 64 90)(3 49 91)(4 50 92)(5 51 93)(6 52 94)(7 53 95)(8 54 96)(9 55 81)(10 56 82)(11 57 83)(12 58 84)(13 59 85)(14 60 86)(15 61 87)(16 62 88)(17 69 35)(18 70 36)(19 71 37)(20 72 38)(21 73 39)(22 74 40)(23 75 41)(24 76 42)(25 77 43)(26 78 44)(27 79 45)(28 80 46)(29 65 47)(30 66 48)(31 67 33)(32 68 34)
(1 65 9 73)(2 66 10 74)(3 67 11 75)(4 68 12 76)(5 69 13 77)(6 70 14 78)(7 71 15 79)(8 72 16 80)(17 85 25 93)(18 86 26 94)(19 87 27 95)(20 88 28 96)(21 89 29 81)(22 90 30 82)(23 91 31 83)(24 92 32 84)(33 57 41 49)(34 58 42 50)(35 59 43 51)(36 60 44 52)(37 61 45 53)(38 62 46 54)(39 63 47 55)(40 64 48 56)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 81)(30 82)(31 83)(32 84)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(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)| (1,63,89)(2,64,90)(3,49,91)(4,50,92)(5,51,93)(6,52,94)(7,53,95)(8,54,96)(9,55,81)(10,56,82)(11,57,83)(12,58,84)(13,59,85)(14,60,86)(15,61,87)(16,62,88)(17,69,35)(18,70,36)(19,71,37)(20,72,38)(21,73,39)(22,74,40)(23,75,41)(24,76,42)(25,77,43)(26,78,44)(27,79,45)(28,80,46)(29,65,47)(30,66,48)(31,67,33)(32,68,34), (1,65,9,73)(2,66,10,74)(3,67,11,75)(4,68,12,76)(5,69,13,77)(6,70,14,78)(7,71,15,79)(8,72,16,80)(17,85,25,93)(18,86,26,94)(19,87,27,95)(20,88,28,96)(21,89,29,81)(22,90,30,82)(23,91,31,83)(24,92,32,84)(33,57,41,49)(34,58,42,50)(35,59,43,51)(36,60,44,52)(37,61,45,53)(38,62,46,54)(39,63,47,55)(40,64,48,56), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,81)(30,82)(31,83)(32,84)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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( (1,63,89)(2,64,90)(3,49,91)(4,50,92)(5,51,93)(6,52,94)(7,53,95)(8,54,96)(9,55,81)(10,56,82)(11,57,83)(12,58,84)(13,59,85)(14,60,86)(15,61,87)(16,62,88)(17,69,35)(18,70,36)(19,71,37)(20,72,38)(21,73,39)(22,74,40)(23,75,41)(24,76,42)(25,77,43)(26,78,44)(27,79,45)(28,80,46)(29,65,47)(30,66,48)(31,67,33)(32,68,34), (1,65,9,73)(2,66,10,74)(3,67,11,75)(4,68,12,76)(5,69,13,77)(6,70,14,78)(7,71,15,79)(8,72,16,80)(17,85,25,93)(18,86,26,94)(19,87,27,95)(20,88,28,96)(21,89,29,81)(22,90,30,82)(23,91,31,83)(24,92,32,84)(33,57,41,49)(34,58,42,50)(35,59,43,51)(36,60,44,52)(37,61,45,53)(38,62,46,54)(39,63,47,55)(40,64,48,56), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,81)(30,82)(31,83)(32,84)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (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([(1,63,89),(2,64,90),(3,49,91),(4,50,92),(5,51,93),(6,52,94),(7,53,95),(8,54,96),(9,55,81),(10,56,82),(11,57,83),(12,58,84),(13,59,85),(14,60,86),(15,61,87),(16,62,88),(17,69,35),(18,70,36),(19,71,37),(20,72,38),(21,73,39),(22,74,40),(23,75,41),(24,76,42),(25,77,43),(26,78,44),(27,79,45),(28,80,46),(29,65,47),(30,66,48),(31,67,33),(32,68,34)], [(1,65,9,73),(2,66,10,74),(3,67,11,75),(4,68,12,76),(5,69,13,77),(6,70,14,78),(7,71,15,79),(8,72,16,80),(17,85,25,93),(18,86,26,94),(19,87,27,95),(20,88,28,96),(21,89,29,81),(22,90,30,82),(23,91,31,83),(24,92,32,84),(33,57,41,49),(34,58,42,50),(35,59,43,51),(36,60,44,52),(37,61,45,53),(38,62,46,54),(39,63,47,55),(40,64,48,56)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,81),(30,82),(31,83),(32,84),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(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 2A2B2C2D3A3B4A4B4C4D4E6A6B6C···6H8A8B8C8D8E···8J12A12B12C12D12E···12J16A···16H16I···16T24A···24H24I···24T48A···48P48Q···48AN
order122223344444666···688888···81212121212···1216···1616···1624···2424···2448···4848···48
size112221111222112···211112···211112···21···12···21···12···21···12···2

120 irreducible representations

dim111111111111111122
type++++
imageC1C2C2C2C3C4C4C6C6C6C8C8C12C12C24C24D4○C16C3×D4○C16
kernelC3×D4○C16C2×C48C3×M5(2)C3×C8○D4D4○C16C3×M4(2)C3×C4○D4C2×C16M5(2)C8○D4C3×D4C3×Q8M4(2)C4○D4D4Q8C3C1
# reps1331262662124124248816

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

3500
010
001
,
100
07980
0218
,
9600
01816
09579
,
9600
0700
0070
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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