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## G = C3⋊SD64order 192 = 26·3

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

Aliases: C33SD64, Q321S3, C12.7D8, C16.6D6, D48.2C2, C24.11D4, C6.10D16, C48.4C22, C3⋊C323C2, (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 >

Character table of C3⋊SD64

 class 1 2A 2B 3 4A 4B 6 8A 8B 12A 12B 12C 16A 16B 16C 16D 24A 24B 32A 32B 32C 32D 32E 32F 32G 32H 48A 48B 48C 48D size 1 1 48 2 2 16 2 2 2 4 16 16 2 2 2 2 4 4 6 6 6 6 6 6 6 6 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 -1 2 -2 -1 2 2 -1 1 1 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ6 2 2 0 -1 2 2 -1 2 2 -1 -1 -1 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 0 2 2 0 2 2 2 2 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ8 2 2 0 2 -2 0 2 0 0 -2 0 0 √2 -√2 √2 -√2 0 0 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 ζ165-ζ163 ζ167-ζ16 ζ167-ζ16 -ζ167+ζ16 -ζ165+ζ163 -√2 √2 -√2 √2 orthogonal lifted from D16 ρ9 2 2 0 2 -2 0 2 0 0 -2 0 0 -√2 √2 -√2 √2 0 0 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 ζ167-ζ16 -ζ165+ζ163 -ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 √2 -√2 √2 -√2 orthogonal lifted from D16 ρ10 2 2 0 2 2 0 2 -2 -2 2 0 0 0 0 0 0 -2 -2 -√2 √2 √2 √2 -√2 -√2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ11 2 2 0 2 2 0 2 -2 -2 2 0 0 0 0 0 0 -2 -2 √2 -√2 -√2 -√2 √2 √2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 2 -2 0 2 0 0 -2 0 0 -√2 √2 -√2 √2 0 0 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 -ζ167+ζ16 ζ165-ζ163 ζ165-ζ163 -ζ165+ζ163 ζ167-ζ16 √2 -√2 √2 -√2 orthogonal lifted from D16 ρ13 2 2 0 2 -2 0 2 0 0 -2 0 0 √2 -√2 √2 -√2 0 0 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ165+ζ163 -ζ167+ζ16 -ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 -√2 √2 -√2 √2 orthogonal lifted from D16 ρ14 2 2 0 -1 2 0 -1 2 2 -1 -√-3 √-3 -2 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 -1 2 0 -1 2 2 -1 √-3 -√-3 -2 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ16 2 -2 0 2 0 0 -2 -√2 √2 0 0 0 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -√2 √2 ζ329+ζ327 ζ3227+ζ3221 ζ3213+ζ323 ζ3229+ζ3219 ζ3231+ζ3217 ζ3215+ζ32 ζ3225+ζ3223 ζ3211+ζ325 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 complex lifted from SD64 ρ17 2 -2 0 2 0 0 -2 -√2 √2 0 0 0 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 -√2 √2 ζ3215+ζ32 ζ3213+ζ323 ζ3211+ζ325 ζ3227+ζ3221 ζ329+ζ327 ζ3225+ζ3223 ζ3231+ζ3217 ζ3229+ζ3219 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 complex lifted from SD64 ρ18 2 -2 0 2 0 0 -2 -√2 √2 0 0 0 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 -√2 √2 ζ3231+ζ3217 ζ3229+ζ3219 ζ3227+ζ3221 ζ3211+ζ325 ζ3225+ζ3223 ζ329+ζ327 ζ3215+ζ32 ζ3213+ζ323 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 complex lifted from SD64 ρ19 2 -2 0 2 0 0 -2 √2 -√2 0 0 0 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 √2 -√2 ζ3211+ζ325 ζ3215+ζ32 ζ3225+ζ3223 ζ329+ζ327 ζ3213+ζ323 ζ3229+ζ3219 ζ3227+ζ3221 ζ3231+ζ3217 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 complex lifted from SD64 ρ20 2 -2 0 2 0 0 -2 √2 -√2 0 0 0 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 √2 -√2 ζ3213+ζ323 ζ329+ζ327 ζ3215+ζ32 ζ3231+ζ3217 ζ3227+ζ3221 ζ3211+ζ325 ζ3229+ζ3219 ζ3225+ζ3223 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 complex lifted from SD64 ρ21 2 -2 0 2 0 0 -2 -√2 √2 0 0 0 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -√2 √2 ζ3225+ζ3223 ζ3211+ζ325 ζ3229+ζ3219 ζ3213+ζ323 ζ3215+ζ32 ζ3231+ζ3217 ζ329+ζ327 ζ3227+ζ3221 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 complex lifted from SD64 ρ22 2 -2 0 2 0 0 -2 √2 -√2 0 0 0 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 √2 -√2 ζ3229+ζ3219 ζ3225+ζ3223 ζ3231+ζ3217 ζ3215+ζ32 ζ3211+ζ325 ζ3227+ζ3221 ζ3213+ζ323 ζ329+ζ327 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 -ζ3210+ζ326 complex lifted from SD64 ρ23 2 -2 0 2 0 0 -2 √2 -√2 0 0 0 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 ζ3214-ζ322 √2 -√2 ζ3227+ζ3221 ζ3231+ζ3217 ζ329+ζ327 ζ3225+ζ3223 ζ3229+ζ3219 ζ3213+ζ323 ζ3211+ζ325 ζ3215+ζ32 ζ3214-ζ322 -ζ3210+ζ326 -ζ3214+ζ322 ζ3210-ζ326 complex lifted from SD64 ρ24 4 4 0 -2 4 0 -2 -4 -4 -2 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 4 0 -2 -4 0 -2 0 0 2 0 0 2√2 -2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from C3⋊D16, Schur index 2 ρ26 4 4 0 -2 -4 0 -2 0 0 2 0 0 -2√2 2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from C3⋊D16, Schur index 2 ρ27 4 -4 0 -2 0 0 2 2√2 -2√2 0 0 0 -2ζ165+2ζ163 2ζ1615-2ζ169 2ζ165-2ζ163 -2ζ1615+2ζ169 -√2 √2 0 0 0 0 0 0 0 0 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 -ζ165+ζ163 orthogonal faithful, Schur index 2 ρ28 4 -4 0 -2 0 0 2 -2√2 2√2 0 0 0 -2ζ1615+2ζ169 -2ζ165+2ζ163 2ζ1615-2ζ169 2ζ165-2ζ163 √2 -√2 0 0 0 0 0 0 0 0 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 ζ167-ζ16 orthogonal faithful, Schur index 2 ρ29 4 -4 0 -2 0 0 2 2√2 -2√2 0 0 0 2ζ165-2ζ163 -2ζ1615+2ζ169 -2ζ165+2ζ163 2ζ1615-2ζ169 -√2 √2 0 0 0 0 0 0 0 0 ζ167-ζ16 -ζ165+ζ163 -ζ167+ζ16 ζ165-ζ163 orthogonal faithful, Schur index 2 ρ30 4 -4 0 -2 0 0 2 -2√2 2√2 0 0 0 2ζ1615-2ζ169 2ζ165-2ζ163 -2ζ1615+2ζ169 -2ζ165+2ζ163 √2 -√2 0 0 0 0 0 0 0 0 ζ165-ζ163 ζ167-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

 1 0 0 0 0 1 0 0 0 0 96 1 0 0 96 0
,
 92 89 0 0 36 19 0 0 0 0 0 1 0 0 1 0
,
 1 33 0 0 0 96 0 0 0 0 0 1 0 0 1 0
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

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