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## G = C3×Q8.D4order 192 = 26·3

### Direct product of C3 and Q8.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×Q8.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×D4 — C3×C4.4D4 — C3×Q8.D4
 Lower central C1 — C2 — C2×C4 — C3×Q8.D4
 Upper central C1 — C2×C6 — C4×C12 — C3×Q8.D4

Generators and relations for C3×Q8.D4
G = < a,b,c,d,e | a3=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 202 in 112 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C3×Q16, C6×D4, C6×Q8, Q8.D4, C3×D4⋊C4, C3×Q8⋊C4, C3×C4⋊C8, Q8×C12, C3×C4.4D4, C6×SD16, C6×Q16, C3×Q8.D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C4○D8, C8.C22, C6×D4, C3×C4○D4, Q8.D4, C3×C4⋊D4, C3×C4○D8, C3×C8.C22, C3×Q8.D4

Smallest permutation representation of C3×Q8.D4
On 96 points
Generators in S96
(1 19 11)(2 20 12)(3 17 9)(4 18 10)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 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 27 3 25)(2 26 4 28)(5 29 7 31)(6 32 8 30)(9 33 11 35)(10 36 12 34)(13 37 15 39)(14 40 16 38)(17 41 19 43)(18 44 20 42)(21 45 23 47)(22 48 24 46)(49 73 51 75)(50 76 52 74)(53 77 55 79)(54 80 56 78)(57 81 59 83)(58 84 60 82)(61 85 63 87)(62 88 64 86)(65 89 67 91)(66 92 68 90)(69 93 71 95)(70 96 72 94)
(1 53 5 51)(2 54 6 52)(3 55 7 49)(4 56 8 50)(9 63 15 57)(10 64 16 58)(11 61 13 59)(12 62 14 60)(17 71 23 65)(18 72 24 66)(19 69 21 67)(20 70 22 68)(25 79 31 73)(26 80 32 74)(27 77 29 75)(28 78 30 76)(33 87 39 81)(34 88 40 82)(35 85 37 83)(36 86 38 84)(41 95 47 89)(42 96 48 90)(43 93 45 91)(44 94 46 92)
(1 49 3 51)(2 52 4 50)(5 55 7 53)(6 54 8 56)(9 59 11 57)(10 58 12 60)(13 63 15 61)(14 62 16 64)(17 67 19 65)(18 66 20 68)(21 71 23 69)(22 70 24 72)(25 76 27 74)(26 75 28 73)(29 80 31 78)(30 79 32 77)(33 84 35 82)(34 83 36 81)(37 88 39 86)(38 87 40 85)(41 92 43 90)(42 91 44 89)(45 96 47 94)(46 95 48 93)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E ··· 4I 4J 6A ··· 6F 6G 6H 8A 8B 8C 8D 12A ··· 12H 12I ··· 12R 12S 12T 24A ··· 24H order 1 2 2 2 2 3 3 4 4 4 4 4 ··· 4 4 6 ··· 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 12 12 24 ··· 24 size 1 1 1 1 8 1 1 2 2 2 2 4 ··· 4 8 1 ··· 1 8 8 4 4 4 4 2 ··· 2 4 ··· 4 8 8 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 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 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 D4 D4 C4○D4 C3×D4 C3×D4 C4○D8 C3×C4○D4 C3×C4○D8 C8.C22 C3×C8.C22 kernel C3×Q8.D4 C3×D4⋊C4 C3×Q8⋊C4 C3×C4⋊C8 Q8×C12 C3×C4.4D4 C6×SD16 C6×Q16 Q8.D4 D4⋊C4 Q8⋊C4 C4⋊C8 C4×Q8 C4.4D4 C2×SD16 C2×Q16 C2×C12 C3×Q8 C12 C2×C4 Q8 C6 C4 C2 C6 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 8 1 2

Matrix representation of C3×Q8.D4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 0
,
 1 0 0 0 71 72 0 0 0 0 6 6 0 0 6 67
,
 46 0 0 0 54 27 0 0 0 0 27 0 0 0 0 27
,
 27 27 0 0 19 46 0 0 0 0 27 0 0 0 0 46
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[1,71,0,0,0,72,0,0,0,0,6,6,0,0,6,67],[46,54,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[27,19,0,0,27,46,0,0,0,0,27,0,0,0,0,46] >;

C3×Q8.D4 in GAP, Magma, Sage, TeX

C_3\times Q_8.D_4
% in TeX

G:=Group("C3xQ8.D4");
// GroupNames label

G:=SmallGroup(192,897);
// by ID

G=gap.SmallGroup(192,897);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,520,6053,1531,124]);
// Polycyclic

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

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