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G = C2×C8.6D6order 192 = 26·3

Direct product of C2 and C8.6D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C8.6D6, Q166D6, C63SD32, C24.24D4, C12.23D8, C24.26C23, D24.9C22, C34(C2×SD32), C3⋊C169C22, (C2×Q16)⋊1S3, (C6×Q16)⋊4C2, (C2×C6).44D8, C6.65(C2×D8), (C2×C8).239D6, C4.10(D4⋊S3), (C2×D24).10C2, (C2×C12).182D4, C12.181(C2×D4), C8.16(C3⋊D4), C8.32(C22×S3), (C3×Q16)⋊6C22, (C2×C24).91C22, C22.23(D4⋊S3), (C2×C3⋊C16)⋊8C2, C2.20(C2×D4⋊S3), C4.11(C2×C3⋊D4), (C2×C4).144(C3⋊D4), SmallGroup(192,737)

Series: Derived Chief Lower central Upper central

C1C24 — C2×C8.6D6
C1C3C6C12C24D24C2×D24 — C2×C8.6D6
C3C6C12C24 — C2×C8.6D6
C1C22C2×C4C2×C8C2×Q16

Generators and relations for C2×C8.6D6
 G = < a,b,c,d | a2=b8=1, c6=b4, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 344 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C16, C2×C8, D8, Q16, Q16, C2×D4, C2×Q8, C24, D12, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C16, SD32, C2×D8, C2×Q16, C3⋊C16, D24, D24, C2×C24, C3×Q16, C3×Q16, C2×D12, C6×Q8, C2×SD32, C2×C3⋊C16, C8.6D6, C2×D24, C6×Q16, C2×C8.6D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, SD32, C2×D8, D4⋊S3, C2×C3⋊D4, C2×SD32, C8.6D6, C2×D4⋊S3, C2×C8.6D6

Smallest permutation representation of C2×C8.6D6
On 96 points
Generators in S96
(1 92)(2 93)(3 94)(4 95)(5 96)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 44)(18 45)(19 46)(20 47)(21 48)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 65)
(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)(49 55 61 51 57 63 53 59)(50 56 62 52 58 64 54 60)(65 71 77 67 73 79 69 75)(66 72 78 68 74 80 70 76)(81 87 93 83 89 95 85 91)(82 88 94 84 90 96 86 92)
(1 59 39 40 58 2 9 51 47 48 50 10)(3 57 41 38 60 16 11 49 33 46 52 8)(4 7 53 45 34 64 12 15 61 37 42 56)(5 55 43 36 62 14 13 63 35 44 54 6)(17 71 81 96 72 32 25 79 89 88 80 24)(18 23 65 87 90 78 26 31 73 95 82 70)(19 69 83 94 74 30 27 77 91 86 66 22)(20 21 67 85 92 76 28 29 75 93 84 68)
(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)| (1,92)(2,93)(3,94)(4,95)(5,96)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,44)(18,45)(19,46)(20,47)(21,48)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,65), (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44)(49,55,61,51,57,63,53,59)(50,56,62,52,58,64,54,60)(65,71,77,67,73,79,69,75)(66,72,78,68,74,80,70,76)(81,87,93,83,89,95,85,91)(82,88,94,84,90,96,86,92), (1,59,39,40,58,2,9,51,47,48,50,10)(3,57,41,38,60,16,11,49,33,46,52,8)(4,7,53,45,34,64,12,15,61,37,42,56)(5,55,43,36,62,14,13,63,35,44,54,6)(17,71,81,96,72,32,25,79,89,88,80,24)(18,23,65,87,90,78,26,31,73,95,82,70)(19,69,83,94,74,30,27,77,91,86,66,22)(20,21,67,85,92,76,28,29,75,93,84,68), (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( (1,92)(2,93)(3,94)(4,95)(5,96)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,44)(18,45)(19,46)(20,47)(21,48)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,65), (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44)(49,55,61,51,57,63,53,59)(50,56,62,52,58,64,54,60)(65,71,77,67,73,79,69,75)(66,72,78,68,74,80,70,76)(81,87,93,83,89,95,85,91)(82,88,94,84,90,96,86,92), (1,59,39,40,58,2,9,51,47,48,50,10)(3,57,41,38,60,16,11,49,33,46,52,8)(4,7,53,45,34,64,12,15,61,37,42,56)(5,55,43,36,62,14,13,63,35,44,54,6)(17,71,81,96,72,32,25,79,89,88,80,24)(18,23,65,87,90,78,26,31,73,95,82,70)(19,69,83,94,74,30,27,77,91,86,66,22)(20,21,67,85,92,76,28,29,75,93,84,68), (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([[(1,92),(2,93),(3,94),(4,95),(5,96),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,44),(18,45),(19,46),(20,47),(21,48),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,65)], [(1,7,13,3,9,15,5,11),(2,8,14,4,10,16,6,12),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28),(33,39,45,35,41,47,37,43),(34,40,46,36,42,48,38,44),(49,55,61,51,57,63,53,59),(50,56,62,52,58,64,54,60),(65,71,77,67,73,79,69,75),(66,72,78,68,74,80,70,76),(81,87,93,83,89,95,85,91),(82,88,94,84,90,96,86,92)], [(1,59,39,40,58,2,9,51,47,48,50,10),(3,57,41,38,60,16,11,49,33,46,52,8),(4,7,53,45,34,64,12,15,61,37,42,56),(5,55,43,36,62,14,13,63,35,44,54,6),(17,71,81,96,72,32,25,79,89,88,80,24),(18,23,65,87,90,78,26,31,73,95,82,70),(19,69,83,94,74,30,27,77,91,86,66,22),(20,21,67,85,92,76,28,29,75,93,84,68)], [(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)]])

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C8A8B8C8D12A12B12C12D12E12F16A···16H24A24B24C24D
order12222234444666888812121212121216···1624242424
size111124242228822222224488886···64444

36 irreducible representations

dim111112222222222444
type+++++++++++++++
imageC1C2C2C2C2S3D4D4D6D6D8D8C3⋊D4C3⋊D4SD32D4⋊S3D4⋊S3C8.6D6
kernelC2×C8.6D6C2×C3⋊C16C8.6D6C2×D24C6×Q16C2×Q16C24C2×C12C2×C8Q16C12C2×C6C8C2×C4C6C4C22C2
# reps114111111222228114

Matrix representation of C2×C8.6D6 in GL5(𝔽97)

960000
01000
00100
00010
00001
,
10000
096000
009600
000076
0003714
,
960000
0155600
0415600
0004367
0009454
,
10000
0824100
0561500
0006330
0005843

G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,0,37,0,0,0,76,14],[96,0,0,0,0,0,15,41,0,0,0,56,56,0,0,0,0,0,43,94,0,0,0,67,54],[1,0,0,0,0,0,82,56,0,0,0,41,15,0,0,0,0,0,63,58,0,0,0,30,43] >;

C2×C8.6D6 in GAP, Magma, Sage, TeX

C_2\times C_8._6D_6
% in TeX

G:=Group("C2xC8.6D6");
// GroupNames label

G:=SmallGroup(192,737);
// by ID

G=gap.SmallGroup(192,737);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,184,675,185,192,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^6=b^4,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations

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