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## G = C3×D4.5D4order 192 = 26·3

### Direct product of C3 and D4.5D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4.5D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×Q8 — C6×Q16 — C3×D4.5D4
 Lower central C1 — C2 — C2×C4 — C3×D4.5D4
 Upper central C1 — C6 — C2×C12 — C3×D4.5D4

Generators and relations for C3×D4.5D4
G = < a,b,c,d,e | a3=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d3 >

Subgroups: 162 in 100 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C12, C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, D4.5D4, C3×C4.10D4, C3×C8.C4, C3×C8○D4, C6×Q16, C3×C8.C22, C3×D4.5D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C6×D4, C3×C4○D4, D4.5D4, C3×C4⋊D4, C3×D4.5D4

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 2 4 1 1 2 2 4 8 8 1 1 2 2 4 4 2 2 4 4 4 8 8 2 2 2 2 4 4 8 8 8 8 2 2 2 2 4 ··· 4 8 8 8 8

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 C4○D4 C3×D4 C3×D4 C3×D4 C3×C4○D4 D4.5D4 C3×D4.5D4 kernel C3×D4.5D4 C3×C4.10D4 C3×C8.C4 C3×C8○D4 C6×Q16 C3×C8.C22 D4.5D4 C4.10D4 C8.C4 C8○D4 C2×Q16 C8.C22 C24 C3×D4 C3×Q8 C2×C6 C8 D4 Q8 C22 C3 C1 # reps 1 2 1 1 1 2 2 4 2 2 2 4 2 1 1 2 4 2 2 4 2 4

Matrix representation of C3×D4.5D4 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 0 1 2 6 4 0 2 1 4 4 1 6 6 1 4 6
,
 3 4 5 0 1 0 3 1 6 6 3 2 0 0 0 1
,
 5 0 2 6 6 5 5 6 6 1 1 2 2 2 6 2
,
 5 0 5 3 5 4 2 2 5 4 0 1 4 5 1 5
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,4,4,6,1,0,4,1,2,2,1,4,6,1,6,6],[3,1,6,0,4,0,6,0,5,3,3,0,0,1,2,1],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[5,5,5,4,0,4,4,5,5,2,0,1,3,2,1,5] >;

C3×D4.5D4 in GAP, Magma, Sage, TeX

C_3\times D_4._5D_4
% in TeX

G:=Group("C3xD4.5D4");
// GroupNames label

G:=SmallGroup(192,906);
// by ID

G=gap.SmallGroup(192,906);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094,4204,172,6053,1531,124]);
// Polycyclic

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

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