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G = C8:S3:C4order 192 = 26·3

2nd semidirect product of C8:S3 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8:5(C4xS3), C8:S3:2C4, C24:10(C2xC4), (C2xC8).65D6, C2.D8:10S3, (C4xS3).2Q8, C4.31(S3xQ8), C4:C4.171D6, D6.6(C4:C4), C8:Dic3:19C2, C12.22(C2xQ8), C6.Q16:20C2, C2.5(D8:S3), C22.91(S3xD4), C6.40(C8:C22), Dic3.7(C4:C4), C12.50(C22xC4), C12.Q8:20C2, C2.5(Q16:S3), (C22xS3).84D4, C3:2(M4(2):C4), (C2xC24).143C22, (C2xC12).297C23, (C2xDic3).167D4, C6.69(C8.C22), C4:Dic3.123C22, C3:C8:5(C2xC4), C4.81(S3xC2xC4), C6.15(C2xC4:C4), (S3xC4:C4).8C2, C2.16(S3xC4:C4), (C3xC2.D8):7C2, (C4xS3).7(C2xC4), C4:C4:7S3.8C2, (C2xC8:S3).4C2, (C2xC6).302(C2xD4), (C2xC3:C8).68C22, (S3xC2xC4).37C22, (C3xC4:C4).90C22, (C2xC4).400(C22xS3), SmallGroup(192,440)

Series: Derived Chief Lower central Upper central

C1C12 — C8:S3:C4
C1C3C6C2xC6C2xC12S3xC2xC4C2xC8:S3 — C8:S3:C4
C3C6C12 — C8:S3:C4
C1C22C2xC4C2.D8

Generators and relations for C8:S3:C4
 G = < a,b,c,d | a8=b3=c2=d4=1, ab=ba, cac=a5, dad-1=a-1, cbc=b-1, bd=db, cd=dc >

Subgroups: 304 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2xC4, C2xC4, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C3:C8, C24, C4xS3, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C4.Q8, C2.D8, C2.D8, C2xC4:C4, C42:C2, C2xM4(2), C8:S3, C2xC3:C8, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C3xC4:C4, C2xC24, S3xC2xC4, S3xC2xC4, M4(2):C4, C6.Q16, C12.Q8, C8:Dic3, C3xC2.D8, S3xC4:C4, C4:C4:7S3, C2xC8:S3, C8:S3:C4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, C23, D6, C4:C4, C22xC4, C2xD4, C2xQ8, C4xS3, C22xS3, C2xC4:C4, C8:C22, C8.C22, S3xC2xC4, S3xD4, S3xQ8, M4(2):C4, S3xC4:C4, D8:S3, Q16:S3, C8:S3:C4

Smallest permutation representation of C8:S3:C4
On 96 points
Generators in S96
(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 71 82)(2 72 83)(3 65 84)(4 66 85)(5 67 86)(6 68 87)(7 69 88)(8 70 81)(9 40 77)(10 33 78)(11 34 79)(12 35 80)(13 36 73)(14 37 74)(15 38 75)(16 39 76)(17 49 58)(18 50 59)(19 51 60)(20 52 61)(21 53 62)(22 54 63)(23 55 64)(24 56 57)(25 47 89)(26 48 90)(27 41 91)(28 42 92)(29 43 93)(30 44 94)(31 45 95)(32 46 96)
(2 6)(4 8)(9 77)(10 74)(11 79)(12 76)(13 73)(14 78)(15 75)(16 80)(17 62)(18 59)(19 64)(20 61)(21 58)(22 63)(23 60)(24 57)(25 89)(26 94)(27 91)(28 96)(29 93)(30 90)(31 95)(32 92)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(65 84)(66 81)(67 86)(68 83)(69 88)(70 85)(71 82)(72 87)
(1 52 45 40)(2 51 46 39)(3 50 47 38)(4 49 48 37)(5 56 41 36)(6 55 42 35)(7 54 43 34)(8 53 44 33)(9 82 20 31)(10 81 21 30)(11 88 22 29)(12 87 23 28)(13 86 24 27)(14 85 17 26)(15 84 18 25)(16 83 19 32)(57 91 73 67)(58 90 74 66)(59 89 75 65)(60 96 76 72)(61 95 77 71)(62 94 78 70)(63 93 79 69)(64 92 80 68)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C8A8B8C8D12A12B12C12D12E12F24A24B24C24D
order1222223444444444444666888812121212121224242424
size111166222444466121212122224412124488884444

36 irreducible representations

dim1111111112222222444444
type+++++++++-+++++--+
imageC1C2C2C2C2C2C2C2C4S3Q8D4D4D6D6C4xS3C8:C22C8.C22S3xQ8S3xD4D8:S3Q16:S3
kernelC8:S3:C4C6.Q16C12.Q8C8:Dic3C3xC2.D8S3xC4:C4C4:C4:7S3C2xC8:S3C8:S3C2.D8C4xS3C2xDic3C22xS3C4:C4C2xC8C8C6C6C4C22C2C2
# reps1111111181211214111122

Matrix representation of C8:S3:C4 in GL8(F73)

720000000
072000000
0045720000
0055280000
0000397030
0000727003
0000312343
0000366913
,
721000000
720000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
007200000
000720000
00001000
00000100
0000472720
0000252072
,
10000000
01000000
0041500000
0016320000
000060700
000071300
00006848713
000028581366

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,45,55,0,0,0,0,0,0,72,28,0,0,0,0,0,0,0,0,39,72,3,36,0,0,0,0,70,70,12,69,0,0,0,0,3,0,34,1,0,0,0,0,0,3,3,3],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,47,25,0,0,0,0,0,1,2,2,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,41,16,0,0,0,0,0,0,50,32,0,0,0,0,0,0,0,0,60,7,68,28,0,0,0,0,7,13,48,58,0,0,0,0,0,0,7,13,0,0,0,0,0,0,13,66] >;

C8:S3:C4 in GAP, Magma, Sage, TeX

C_8\rtimes S_3\rtimes C_4
% in TeX

G:=Group("C8:S3:C4");
// GroupNames label

G:=SmallGroup(192,440);
// by ID

G=gap.SmallGroup(192,440);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,219,58,438,102,6278]);
// Polycyclic

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

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