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G = C3×D4.C8order 192 = 26·3

Direct product of C3 and D4.C8

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

Aliases: C3×D4.C8, D4.C24, Q8.2C24, C24.107D4, M5(2)⋊4C6, M4(2).3C12, (C2×C48)⋊6C2, (C2×C16)⋊2C6, C4.3(C2×C24), C8○D4.4C6, (C3×D4).3C8, C8.27(C3×D4), (C3×Q8).3C8, C12.32(C2×C8), C4○D4.4C12, (C2×C6).8M4(2), C6.27(C22⋊C8), (C3×M5(2))⋊12C2, (C3×M4(2)).7C4, (C2×C24).442C22, C12.113(C22⋊C4), C22.1(C3×M4(2)), (C2×C8).96(C2×C6), (C3×C4○D4).5C4, (C3×C8○D4).5C2, C2.8(C3×C22⋊C8), (C2×C4).42(C2×C12), C4.30(C3×C22⋊C4), (C2×C12).263(C2×C4), SmallGroup(192,156)

Series: Derived Chief Lower central Upper central

C1C4 — C3×D4.C8
C1C2C4C8C2×C8C2×C24C2×C48 — C3×D4.C8
C1C2C4 — C3×D4.C8
C1C24C2×C24 — C3×D4.C8

Generators and relations for C3×D4.C8
 G = < a,b,c,d | a3=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >

2C2
4C2
2C4
2C22
2C6
4C6
2C2×C4
2D4
2C8
2C2×C6
2C12
2C2×C8
2C16
2M4(2)
2C16
2C3×D4
2C2×C12
2C24
2C3×M4(2)
2C48
2C48
2C2×C24

Smallest permutation representation of C3×D4.C8
On 96 points
Generators in S96
(1 57 74)(2 58 75)(3 59 76)(4 60 77)(5 61 78)(6 62 79)(7 63 80)(8 64 65)(9 49 66)(10 50 67)(11 51 68)(12 52 69)(13 53 70)(14 54 71)(15 55 72)(16 56 73)(17 88 48)(18 89 33)(19 90 34)(20 91 35)(21 92 36)(22 93 37)(23 94 38)(24 95 39)(25 96 40)(26 81 41)(27 82 42)(28 83 43)(29 84 44)(30 85 45)(31 86 46)(32 87 47)
(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 64 25 56)(18 49 26 57)(19 50 27 58)(20 51 28 59)(21 52 29 60)(22 53 30 61)(23 54 31 62)(24 55 32 63)(65 96 73 88)(66 81 74 89)(67 82 75 90)(68 83 76 91)(69 84 77 92)(70 85 78 93)(71 86 79 94)(72 87 80 95)
(1 41)(2 10)(3 35)(5 45)(6 14)(7 39)(9 33)(11 43)(13 37)(15 47)(17 25)(18 49)(20 59)(21 29)(22 53)(24 63)(26 57)(28 51)(30 61)(32 55)(36 44)(40 48)(50 58)(54 62)(66 89)(67 75)(68 83)(70 93)(71 79)(72 87)(74 81)(76 91)(78 85)(80 95)(84 92)(88 96)
(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)| (1,57,74)(2,58,75)(3,59,76)(4,60,77)(5,61,78)(6,62,79)(7,63,80)(8,64,65)(9,49,66)(10,50,67)(11,51,68)(12,52,69)(13,53,70)(14,54,71)(15,55,72)(16,56,73)(17,88,48)(18,89,33)(19,90,34)(20,91,35)(21,92,36)(22,93,37)(23,94,38)(24,95,39)(25,96,40)(26,81,41)(27,82,42)(28,83,43)(29,84,44)(30,85,45)(31,86,46)(32,87,47), (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)(17,64,25,56)(18,49,26,57)(19,50,27,58)(20,51,28,59)(21,52,29,60)(22,53,30,61)(23,54,31,62)(24,55,32,63)(65,96,73,88)(66,81,74,89)(67,82,75,90)(68,83,76,91)(69,84,77,92)(70,85,78,93)(71,86,79,94)(72,87,80,95), (1,41)(2,10)(3,35)(5,45)(6,14)(7,39)(9,33)(11,43)(13,37)(15,47)(17,25)(18,49)(20,59)(21,29)(22,53)(24,63)(26,57)(28,51)(30,61)(32,55)(36,44)(40,48)(50,58)(54,62)(66,89)(67,75)(68,83)(70,93)(71,79)(72,87)(74,81)(76,91)(78,85)(80,95)(84,92)(88,96), (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( (1,57,74)(2,58,75)(3,59,76)(4,60,77)(5,61,78)(6,62,79)(7,63,80)(8,64,65)(9,49,66)(10,50,67)(11,51,68)(12,52,69)(13,53,70)(14,54,71)(15,55,72)(16,56,73)(17,88,48)(18,89,33)(19,90,34)(20,91,35)(21,92,36)(22,93,37)(23,94,38)(24,95,39)(25,96,40)(26,81,41)(27,82,42)(28,83,43)(29,84,44)(30,85,45)(31,86,46)(32,87,47), (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)(17,64,25,56)(18,49,26,57)(19,50,27,58)(20,51,28,59)(21,52,29,60)(22,53,30,61)(23,54,31,62)(24,55,32,63)(65,96,73,88)(66,81,74,89)(67,82,75,90)(68,83,76,91)(69,84,77,92)(70,85,78,93)(71,86,79,94)(72,87,80,95), (1,41)(2,10)(3,35)(5,45)(6,14)(7,39)(9,33)(11,43)(13,37)(15,47)(17,25)(18,49)(20,59)(21,29)(22,53)(24,63)(26,57)(28,51)(30,61)(32,55)(36,44)(40,48)(50,58)(54,62)(66,89)(67,75)(68,83)(70,93)(71,79)(72,87)(74,81)(76,91)(78,85)(80,95)(84,92)(88,96), (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([(1,57,74),(2,58,75),(3,59,76),(4,60,77),(5,61,78),(6,62,79),(7,63,80),(8,64,65),(9,49,66),(10,50,67),(11,51,68),(12,52,69),(13,53,70),(14,54,71),(15,55,72),(16,56,73),(17,88,48),(18,89,33),(19,90,34),(20,91,35),(21,92,36),(22,93,37),(23,94,38),(24,95,39),(25,96,40),(26,81,41),(27,82,42),(28,83,43),(29,84,44),(30,85,45),(31,86,46),(32,87,47)], [(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),(17,64,25,56),(18,49,26,57),(19,50,27,58),(20,51,28,59),(21,52,29,60),(22,53,30,61),(23,54,31,62),(24,55,32,63),(65,96,73,88),(66,81,74,89),(67,82,75,90),(68,83,76,91),(69,84,77,92),(70,85,78,93),(71,86,79,94),(72,87,80,95)], [(1,41),(2,10),(3,35),(5,45),(6,14),(7,39),(9,33),(11,43),(13,37),(15,47),(17,25),(18,49),(20,59),(21,29),(22,53),(24,63),(26,57),(28,51),(30,61),(32,55),(36,44),(40,48),(50,58),(54,62),(66,89),(67,75),(68,83),(70,93),(71,79),(72,87),(74,81),(76,91),(78,85),(80,95),(84,92),(88,96)], [(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)])

84 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H16A···16H16I16J16K16L24A···24H24I24J24K24L24M24N24O24P48A···48P48Q···48X
order122233444466666688888888121212121212121216···161616161624···24242424242424242448···4848···48
size112411112411224411112244111122442···244441···1222244442···24···4

84 irreducible representations

dim1111111111111111222222
type+++++
imageC1C2C2C2C3C4C4C6C6C6C8C8C12C12C24C24D4M4(2)C3×D4C3×M4(2)D4.C8C3×D4.C8
kernelC3×D4.C8C2×C48C3×M5(2)C3×C8○D4D4.C8C3×M4(2)C3×C4○D4C2×C16M5(2)C8○D4C3×D4C3×Q8M4(2)C4○D4D4Q8C24C2×C6C8C22C3C1
# reps11112222224444882244816

Matrix representation of C3×D4.C8 in GL3(𝔽97) generated by

3500
010
001
,
100
0033
0470
,
9600
0033
0500
,
2200
0568
0415
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[1,0,0,0,0,47,0,33,0],[96,0,0,0,0,50,0,33,0],[22,0,0,0,5,41,0,68,5] >;

C3×D4.C8 in GAP, Magma, Sage, TeX

C_3\times D_4.C_8
% in TeX

G:=Group("C3xD4.C8");
// GroupNames label

G:=SmallGroup(192,156);
// by ID

G=gap.SmallGroup(192,156);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1522,248,2111,102,124]);
// 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,d*c*d^-1=b*c>;
// generators/relations

Export

Subgroup lattice of C3×D4.C8 in TeX

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