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## G = C3⋊C8.6D4order 192 = 26·3

### 6th non-split extension by C3⋊C8 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C3⋊C8.6D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — C3⋊C8.6D4
 Lower central C3 — C6 — C2×C12 — C3⋊C8.6D4
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for C3⋊C8.6D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, dbd=b5, dcd=b4c-1 >

Subgroups: 272 in 110 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C2×Q8, C3⋊C8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, Q8⋊C4, C4.Q8, C22⋊Q8, C22⋊Q8, C2×M4(2), C2×Q16, C2×C3⋊C8, C4.Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C8.D4, C12.Q8, C6.SD16, Q82Dic3, C2×C4.Dic3, C12.48D4, C2×C3⋊Q16, C3×C22⋊Q8, C3⋊C8.6D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C8.C22, S3×D4, D42S3, C2×C3⋊D4, C8.D4, C23.14D6, Q8.11D6, Q8.14D6, C3⋊C8.6D4

Character table of C3⋊C8.6D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 4 2 2 2 4 8 8 24 24 2 2 2 4 4 12 12 12 12 4 4 4 4 8 8 8 8 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 -2 -2 0 2 -2 2 0 0 0 0 0 2 -2 -2 0 0 0 -2 2 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -1 2 2 -2 -2 2 0 0 -1 -1 -1 1 1 0 0 0 0 -1 1 -1 1 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 -1 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -2 -1 2 2 -2 2 -2 0 0 -1 -1 -1 1 1 0 0 0 0 -1 1 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ13 2 2 2 2 2 -1 2 2 2 -2 -2 0 0 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ14 2 2 2 2 -2 2 -2 -2 2 0 0 0 0 2 2 2 -2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 2 -2 2 0 0 0 0 0 2 -2 -2 0 0 0 2 -2 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 2 2 -2 -2 -2 0 0 0 0 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 2 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 0 0 0 0 1 -1 1 -1 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ19 2 2 2 2 2 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ20 2 2 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 0 0 0 0 1 -1 1 -1 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 2i 0 0 -2i -2 0 2 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 -2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 -2i 0 0 2i -2 0 2 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 -2 -4 4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 -2 0 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 4 -4 0 4 0 0 0 0 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ25 4 -4 -4 4 0 4 0 0 0 0 0 0 0 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ26 4 4 -4 -4 0 -2 4 -4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ27 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 2√3 0 -2√3 0 0 0 0 symplectic lifted from Q8.14D6, Schur index 2 ρ28 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 -2√3 0 2√3 0 0 0 0 symplectic lifted from Q8.14D6, Schur index 2 ρ29 4 -4 -4 4 0 -2 0 0 0 0 0 0 0 2 -2 2 2√-3 -2√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from Q8.11D6 ρ30 4 -4 -4 4 0 -2 0 0 0 0 0 0 0 2 -2 2 -2√-3 2√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from Q8.11D6

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

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

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

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

Matrix representation of C3⋊C8.6D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 62 25 0 0 0 0 36 11 0 0 11 36 0 0 0 0 25 62 0 0
,
 34 3 0 0 0 0 28 39 0 0 0 0 0 0 62 37 0 0 0 0 48 11 0 0 0 0 0 0 11 48 0 0 0 0 37 62
,
 72 0 0 0 0 0 47 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,25,0,0,0,0,36,62,0,0,62,36,0,0,0,0,25,11,0,0],[34,28,0,0,0,0,3,39,0,0,0,0,0,0,62,48,0,0,0,0,37,11,0,0,0,0,0,0,11,37,0,0,0,0,48,62],[72,47,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C3⋊C8.6D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8._6D_4
% in TeX

G:=Group("C3:C8.6D4");
// GroupNames label

G:=SmallGroup(192,611);
// by ID

G=gap.SmallGroup(192,611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,555,184,1123,297,136,6278]);
// Polycyclic

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

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