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G = C3⋊SD64  order 192 = 26·3

The semidirect product of C3 and SD64 acting via SD64/Q32=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊3SD64, Q32⋊1S3, C12.7D8, C16.6D6, D48.2C2, C24.11D4, C6.10D16, C48.4C22, C3⋊C32⋊3C2, (C3×Q32)⋊1C2, C4.3(D4⋊S3), C8.11(C3⋊D4), C2.6(C3⋊D16), SmallGroup(192,80)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C48 — C3⋊SD64
Chief series
C1 — C3 — C6 — C12 — C24 — C48 — D48 — C3⋊SD64
Lower central
C3 — C6 — C12 — C24 — C48 — C3⋊SD64
Upper central
C1 — C2 — C4 — C8 — C16 — Q32

Generators and relations for C3⋊SD64
 G = < a,b,c | a3=b32=c2=1, bab-1=cac=a-1, cbc=b15 >

C1
C2
48C2
C3
C4
8C4
24C22
C6
16S3
C8
4Q8
12D4
C12
8D6
8C12
C16
2Q16
6D8
C24
4D12
4C3×Q8
Q32
3C32
3D16
C48
2D24
2C3×Q16
3SD64
D48
C3⋊C32
C3×Q32
C3⋊SD64

Character table of C3⋊SD64

 class 12A2B34A4B68A8B12A12B12C16A16B16C16D24A24B32A32B32C32D32E32F32G32H48A48B48C48D
 size 1148221622241616222244666666664444
ρ1111111111111111111111111111111    trivial
ρ211111-11111-1-1111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ311-1111111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ411-111-11111-1-1111111111111111111    linear of order 2
ρ5220-12-2-122-1112222-1-100000000-1-1-1-1    orthogonal lifted from D6
ρ6220-122-122-1-1-12222-1-100000000-1-1-1-1    orthogonal lifted from S3
ρ7220220222200-2-2-2-22200000000-2-2-2-2    orthogonal lifted from D4
ρ82202-20200-200√2-√2√2-√200-ζ167+ζ16-ζ165+ζ163ζ165-ζ163ζ165-ζ163ζ167-ζ16ζ167-ζ16-ζ167+ζ16-ζ165+ζ163-√2√2-√2√2    orthogonal lifted from D16
ρ92202-20200-200-√2√2-√2√200ζ165-ζ163-ζ167+ζ16ζ167-ζ16ζ167-ζ16-ζ165+ζ163-ζ165+ζ163ζ165-ζ163-ζ167+ζ16√2-√2√2-√2    orthogonal lifted from D16
ρ102202202-2-22000000-2-2-√2√2√2√2-√2-√2-√2√20000    orthogonal lifted from D8
ρ112202202-2-22000000-2-2√2-√2-√2-√2√2√2√2-√20000    orthogonal lifted from D8
ρ122202-20200-200-√2√2-√2√200-ζ165+ζ163ζ167-ζ16-ζ167+ζ16-ζ167+ζ16ζ165-ζ163ζ165-ζ163-ζ165+ζ163ζ167-ζ16√2-√2√2-√2    orthogonal lifted from D16
ρ132202-20200-200√2-√2√2-√200ζ167-ζ16ζ165-ζ163-ζ165+ζ163-ζ165+ζ163-ζ167+ζ16-ζ167+ζ16ζ167-ζ16ζ165-ζ163-√2√2-√2√2    orthogonal lifted from D16
ρ14220-120-122-1-√-3√-3-2-2-2-2-1-1000000001111    complex lifted from C3⋊D4
ρ15220-120-122-1√-3-√-3-2-2-2-2-1-1000000001111    complex lifted from C3⋊D4
ρ162-20200-2-√2√2000-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326-√2√2ζ329+ζ327ζ3227+ζ3221ζ3213+ζ323ζ3229+ζ3219ζ3231+ζ3217ζ3215+ζ32ζ3225+ζ3223ζ3211+ζ325-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322    complex lifted from SD64
ρ172-20200-2-√2√2000ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326-√2√2ζ3215+ζ32ζ3213+ζ323ζ3211+ζ325ζ3227+ζ3221ζ329+ζ327ζ3225+ζ3223ζ3231+ζ3217ζ3229+ζ3219ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322    complex lifted from SD64
ρ182-20200-2-√2√2000ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326-√2√2ζ3231+ζ3217ζ3229+ζ3219ζ3227+ζ3221ζ3211+ζ325ζ3225+ζ3223ζ329+ζ327ζ3215+ζ32ζ3213+ζ323ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322    complex lifted from SD64
ρ192-20200-2√2-√2000-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322√2-√2ζ3211+ζ325ζ3215+ζ32ζ3225+ζ3223ζ329+ζ327ζ3213+ζ323ζ3229+ζ3219ζ3227+ζ3221ζ3231+ζ3217ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326    complex lifted from SD64
ρ202-20200-2√2-√2000ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322√2-√2ζ3213+ζ323ζ329+ζ327ζ3215+ζ32ζ3231+ζ3217ζ3227+ζ3221ζ3211+ζ325ζ3229+ζ3219ζ3225+ζ3223-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326    complex lifted from SD64
ρ212-20200-2-√2√2000-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326-√2√2ζ3225+ζ3223ζ3211+ζ325ζ3229+ζ3219ζ3213+ζ323ζ3215+ζ32ζ3231+ζ3217ζ329+ζ327ζ3227+ζ3221-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322    complex lifted from SD64
ρ222-20200-2√2-√2000ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322√2-√2ζ3229+ζ3219ζ3225+ζ3223ζ3231+ζ3217ζ3215+ζ32ζ3211+ζ325ζ3227+ζ3221ζ3213+ζ323ζ329+ζ327-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322-ζ3210+ζ326    complex lifted from SD64
ρ232-20200-2√2-√2000-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326ζ3214-ζ322√2-√2ζ3227+ζ3221ζ3231+ζ3217ζ329+ζ327ζ3225+ζ3223ζ3229+ζ3219ζ3213+ζ323ζ3211+ζ325ζ3215+ζ32ζ3214-ζ322-ζ3210+ζ326-ζ3214+ζ322ζ3210-ζ326    complex lifted from SD64
ρ24440-240-2-4-4-200000022000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ25440-2-40-2002002√2-2√22√2-2√20000000000√2-√2√2-√2    orthogonal lifted from C3⋊D16, Schur index 2
ρ26440-2-40-200200-2√22√2-2√22√20000000000-√2√2-√2√2    orthogonal lifted from C3⋊D16, Schur index 2
ρ274-40-20022√2-2√2000-2ζ165+2ζ1632ζ1615-2ζ1692ζ165-2ζ163-2ζ1615+2ζ169-√2√200000000-ζ167+ζ16ζ165-ζ163ζ167-ζ16-ζ165+ζ163    orthogonal faithful, Schur index 2
ρ284-40-2002-2√22√2000-2ζ1615+2ζ169-2ζ165+2ζ1632ζ1615-2ζ1692ζ165-2ζ163√2-√200000000-ζ165+ζ163-ζ167+ζ16ζ165-ζ163ζ167-ζ16    orthogonal faithful, Schur index 2
ρ294-40-20022√2-2√20002ζ165-2ζ163-2ζ1615+2ζ169-2ζ165+2ζ1632ζ1615-2ζ169-√2√200000000ζ167-ζ16-ζ165+ζ163-ζ167+ζ16ζ165-ζ163    orthogonal faithful, Schur index 2
ρ304-40-2002-2√22√20002ζ1615-2ζ1692ζ165-2ζ163-2ζ1615+2ζ169-2ζ165+2ζ163√2-√200000000ζ165-ζ163ζ167-ζ16-ζ165+ζ163-ζ167+ζ16    orthogonal faithful, Schur index 2

Smallest permutation representation of C3⋊SD64
►On 96 points
Generators in S96
(1 75 35)(2 36 76)(3 77 37)(4 38 78)(5 79 39)(6 40 80)(7 81 41)(8 42 82)(9 83 43)(10 44 84)(11 85 45)(12 46 86)(13 87 47)(14 48 88)(15 89 49)(16 50 90)(17 91 51)(18 52 92)(19 93 53)(20 54 94)(21 95 55)(22 56 96)(23 65 57)(24 58 66)(25 67 59)(26 60 68)(27 69 61)(28 62 70)(29 71 63)(30 64 72)(31 73 33)(32 34 74)

(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)

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

Matrix representation of C3⋊SD64 ►in GL4(𝔽97) generated by

1000
0100
00961
00960
,
928900
361900
0001
0010
,
13300
09600
0001
0010
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,96,96,0,0,1,0],[92,36,0,0,89,19,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,33,96,0,0,0,0,0,1,0,0,1,0] >;
 

C3⋊SD64 in GAP, Magma, Sage, TeX 

C_3\rtimes {\rm SD}_{64}
 
% in TeX
 
G:=Group("C3:SD64");
 
// GroupNames label
 
G:=SmallGroup(192,80);
 
// by ID
 
G=gap.SmallGroup(192,80);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,232,254,135,142,675,346,192,1684,851,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^15>;
 
// generators/relations
 

Export

Subgroup lattice of C3⋊SD64 in TeX
Character table of C3⋊SD64 in TeX

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