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## G = C3×D4⋊2Q8order 192 = 26·3

### Direct product of C3 and D4⋊2Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×D4⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C4⋊Q8 — C3×D4⋊2Q8
 Lower central C1 — C2 — C2×C4 — C3×D4⋊2Q8
 Upper central C1 — C2×C6 — C4×C12 — C3×D4⋊2Q8

Generators and relations for C3×D42Q8
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >

Subgroups: 202 in 108 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C6×D4, C6×Q8, D42Q8, C3×D4⋊C4, C3×C4⋊C8, C3×C4.Q8, D4×C12, C3×C4⋊Q8, C3×D42Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, SD16, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C2×SD16, C8⋊C22, C3×SD16, C6×D4, C6×Q8, C3×C4○D4, D42Q8, C3×C22⋊Q8, C6×SD16, C3×C8⋊C22, C3×D42Q8

Smallest permutation representation of C3×D42Q8
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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 22)(18 21)(19 24)(20 23)(25 30)(26 29)(27 32)(28 31)(33 38)(34 37)(35 40)(36 39)(41 46)(42 45)(43 48)(44 47)(49 53)(50 56)(51 55)(52 54)(57 61)(58 64)(59 63)(60 62)(65 69)(66 72)(67 71)(68 70)(73 77)(74 80)(75 79)(76 78)(81 85)(82 88)(83 87)(84 86)(89 93)(90 96)(91 95)(92 94)
(1 32 5 26)(2 29 6 27)(3 30 7 28)(4 31 8 25)(9 38 15 36)(10 39 16 33)(11 40 13 34)(12 37 14 35)(17 46 23 44)(18 47 24 41)(19 48 21 42)(20 45 22 43)(49 74 55 80)(50 75 56 77)(51 76 53 78)(52 73 54 79)(57 82 63 88)(58 83 64 85)(59 84 61 86)(60 81 62 87)(65 90 71 96)(66 91 72 93)(67 92 69 94)(68 89 70 95)
(1 55 5 49)(2 54 6 52)(3 53 7 51)(4 56 8 50)(9 61 15 59)(10 64 16 58)(11 63 13 57)(12 62 14 60)(17 69 23 67)(18 72 24 66)(19 71 21 65)(20 70 22 68)(25 77 31 75)(26 80 32 74)(27 79 29 73)(28 78 30 76)(33 85 39 83)(34 88 40 82)(35 87 37 81)(36 86 38 84)(41 93 47 91)(42 96 48 90)(43 95 45 89)(44 94 46 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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,23)(25,30)(26,29)(27,32)(28,31)(33,38)(34,37)(35,40)(36,39)(41,46)(42,45)(43,48)(44,47)(49,53)(50,56)(51,55)(52,54)(57,61)(58,64)(59,63)(60,62)(65,69)(66,72)(67,71)(68,70)(73,77)(74,80)(75,79)(76,78)(81,85)(82,88)(83,87)(84,86)(89,93)(90,96)(91,95)(92,94), (1,32,5,26)(2,29,6,27)(3,30,7,28)(4,31,8,25)(9,38,15,36)(10,39,16,33)(11,40,13,34)(12,37,14,35)(17,46,23,44)(18,47,24,41)(19,48,21,42)(20,45,22,43)(49,74,55,80)(50,75,56,77)(51,76,53,78)(52,73,54,79)(57,82,63,88)(58,83,64,85)(59,84,61,86)(60,81,62,87)(65,90,71,96)(66,91,72,93)(67,92,69,94)(68,89,70,95), (1,55,5,49)(2,54,6,52)(3,53,7,51)(4,56,8,50)(9,61,15,59)(10,64,16,58)(11,63,13,57)(12,62,14,60)(17,69,23,67)(18,72,24,66)(19,71,21,65)(20,70,22,68)(25,77,31,75)(26,80,32,74)(27,79,29,73)(28,78,30,76)(33,85,39,83)(34,88,40,82)(35,87,37,81)(36,86,38,84)(41,93,47,91)(42,96,48,90)(43,95,45,89)(44,94,46,92)>;

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

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

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

Matrix representation of C3×D42Q8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 0 1 0 0 72 0 0 0 0 0 72 0 0 0 0 72
,
 0 72 0 0 72 0 0 0 0 0 1 0 0 0 0 72
,
 0 1 0 0 72 0 0 0 0 0 27 0 0 0 0 46
,
 6 67 0 0 67 67 0 0 0 0 0 1 0 0 72 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,0,72],[0,72,0,0,1,0,0,0,0,0,27,0,0,0,0,46],[6,67,0,0,67,67,0,0,0,0,0,72,0,0,1,0] >;

C3×D42Q8 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_2Q_8
% in TeX

G:=Group("C3xD4:2Q8");
// GroupNames label

G:=SmallGroup(192,909);
// by ID

G=gap.SmallGroup(192,909);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,840,365,176,1094,6053,1531,124]);
// Polycyclic

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

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