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## G = C3×D4.Q8order 192 = 26·3

### Direct product of C3 and D4.Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C42.C2 — C3×D4.Q8
 Lower central C1 — C2 — C2×C4 — C3×D4.Q8
 Upper central C1 — C2×C6 — C4×C12 — C3×D4.Q8

Generators and relations for C3×D4.Q8
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=b2d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 186 in 102 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C6×D4, D4.Q8, C3×D4⋊C4, C3×C4⋊C8, C3×C4.Q8, C3×C2.D8, D4×C12, C3×C42.C2, C3×D4.Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C4○D8, C8⋊C22, C6×D4, C6×Q8, C3×C4○D4, D4.Q8, C3×C22⋊Q8, C3×C4○D8, C3×C8⋊C22, C3×D4.Q8

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A ··· 12H 12I ··· 12N 12O 12P 12Q 12R 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 4 4 1 1 2 2 2 2 4 4 4 8 8 1 ··· 1 4 4 4 4 4 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

57 irreducible representations

 dim 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 C3 C6 C6 C6 C6 C6 C6 D4 Q8 C4○D4 C3×D4 C3×Q8 C4○D8 C3×C4○D4 C3×C4○D8 C8⋊C22 C3×C8⋊C22 kernel C3×D4.Q8 C3×D4⋊C4 C3×C4⋊C8 C3×C4.Q8 C3×C2.D8 D4×C12 C3×C42.C2 D4.Q8 D4⋊C4 C4⋊C8 C4.Q8 C2.D8 C4×D4 C42.C2 C2×C12 C3×D4 C12 C2×C4 D4 C6 C4 C2 C6 C2 # reps 1 2 1 1 1 1 1 2 4 2 2 2 2 2 2 2 2 4 4 4 4 8 1 2

Matrix representation of C3×D4.Q8 in GL5(𝔽73)

 8 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 72
,
 72 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 46
,
 72 0 0 0 0 0 16 57 0 0 0 57 57 0 0 0 0 0 0 1 0 0 0 72 0

G:=sub<GL(5,GF(73))| [8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72],[72,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,46],[72,0,0,0,0,0,16,57,0,0,0,57,57,0,0,0,0,0,0,72,0,0,0,1,0] >;

C3×D4.Q8 in GAP, Magma, Sage, TeX

C_3\times D_4.Q_8
% in TeX

G:=Group("C3xD4.Q8");
// GroupNames label

G:=SmallGroup(192,911);
// by ID

G=gap.SmallGroup(192,911);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,848,1094,6053,1531,124]);
// Polycyclic

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

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