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G = C3×C8.2D4order 192 = 26·3

Direct product of C3 and C8.2D4

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

Aliases: C3×C8.2D4, C24.47D4, C4⋊Q88C6, C8⋊C45C6, C4.6(C6×D4), C8.2(C3×D4), (C2×Q16)⋊9C6, (C6×Q16)⋊23C2, C12.313(C2×D4), (C2×C12).344D4, C42.30(C2×C6), (C2×SD16).2C6, (C6×SD16).5C2, C4.4D4.7C6, C6.47(C41D4), C22.118(C6×D4), (C4×C12).272C22, (C2×C12).953C23, (C2×C24).204C22, (C6×D4).206C22, (C6×Q8).180C22, C6.147(C8.C22), (C3×C4⋊Q8)⋊29C2, (C2×C8).28(C2×C6), (C3×C8⋊C4)⋊10C2, (C2×C4).45(C3×D4), (C2×D4).29(C2×C6), (C2×C6).674(C2×D4), C2.10(C3×C41D4), (C2×Q8).25(C2×C6), C2.22(C3×C8.C22), (C2×C4).128(C22×C6), (C3×C4.4D4).16C2, SmallGroup(192,930)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C3×C8.2D4
C1C2C22C2×C4C2×C12C6×Q8C6×SD16 — C3×C8.2D4
C1C2C2×C4 — C3×C8.2D4
C1C2×C6C4×C12 — C3×C8.2D4

Generators and relations for C3×C8.2D4
 G = < a,b,c,d | a3=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b3, dcd=c3 >

Subgroups: 226 in 124 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C3×Q16, C6×D4, C6×Q8, C6×Q8, C8.2D4, C3×C8⋊C4, C3×C4.4D4, C3×C4⋊Q8, C6×SD16, C6×Q16, C3×C8.2D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C41D4, C8.C22, C6×D4, C8.2D4, C3×C41D4, C3×C8.C22, C3×C8.2D4

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

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

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

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

48 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E4F4G6A···6F6G6H8A8B8C8D12A12B12C12D12E12F12G12H12I···12N24A···24H
order122223344444446···6668888121212121212121212···1224···24
size111181122448881···1884444222244448···84···4

48 irreducible representations

dim111111111111222244
type++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3×D4C3×D4C8.C22C3×C8.C22
kernelC3×C8.2D4C3×C8⋊C4C3×C4.4D4C3×C4⋊Q8C6×SD16C6×Q16C8.2D4C8⋊C4C4.4D4C4⋊Q8C2×SD16C2×Q16C24C2×C12C8C2×C4C6C2
# reps111122222244428424

Matrix representation of C3×C8.2D4 in GL8(𝔽73)

640000000
064000000
00100000
00010000
00001000
00000100
00000010
00000001
,
171000000
172000000
000720000
00100000
000023235023
000050235050
000023505050
000023232350
,
722000000
721000000
00100000
00010000
000023505050
000023232350
000023235023
000050235050
,
720000000
721000000
00100000
000720000
000007200
000072000
00000001
00000010

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,23,50,23,23,0,0,0,0,23,23,50,23,0,0,0,0,50,50,50,23,0,0,0,0,23,50,50,50],[72,72,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,23,23,50,0,0,0,0,50,23,23,23,0,0,0,0,50,23,50,50,0,0,0,0,50,50,23,50],[72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3×C8.2D4 in GAP, Magma, Sage, TeX

C_3\times C_8._2D_4
% in TeX

G:=Group("C3xC8.2D4");
// GroupNames label

G:=SmallGroup(192,930);
// by ID

G=gap.SmallGroup(192,930);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,176,1094,1059,268,4204,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^3,d*c*d=c^3>;
// generators/relations

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