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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 — C22 — C2×C4 — C2×C12 — C6×D4 — C6×SD16 — C3×Q8⋊D4
 Lower central C1 — C2 — C2×C4 — C3×Q8⋊D4
 Upper central C1 — C2×C6 — C22×C12 — C3×Q8⋊D4

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

Subgroups: 290 in 158 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C6×Q8, Q8⋊D4, C3×C22⋊C8, C3×Q8⋊C4, C3×C4⋊D4, C6×SD16, Q8×C2×C6, C3×Q8⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C3×D4, C22×C6, C22≀C2, C2×SD16, C8.C22, C3×SD16, C6×D4, Q8⋊D4, C3×C22≀C2, C6×SD16, C3×C8.C22, C3×Q8⋊D4

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C ··· 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12N 12O 12P 24A ··· 24H order 1 2 2 2 2 2 2 3 3 4 4 4 ··· 4 4 6 ··· 6 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 2 2 8 1 1 2 2 4 ··· 4 8 1 ··· 1 2 2 2 2 8 8 4 4 4 4 2 2 2 2 4 ··· 4 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 D4 D4 SD16 C3×D4 C3×D4 C3×D4 C3×SD16 C8.C22 C3×C8.C22 kernel C3×Q8⋊D4 C3×C22⋊C8 C3×Q8⋊C4 C3×C4⋊D4 C6×SD16 Q8×C2×C6 Q8⋊D4 C22⋊C8 Q8⋊C4 C4⋊D4 C2×SD16 C22×Q8 C2×C12 C3×Q8 C22×C6 C2×C6 C2×C4 Q8 C23 C22 C6 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 4 1 4 2 8 2 8 1 2

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

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 72 0 0 0 0 72 0 0 0 0 1 71 0 0 1 72
,
 72 71 0 0 0 1 0 0 0 0 61 12 0 0 67 12
,
 1 2 0 0 72 72 0 0 0 0 1 0 0 0 1 72
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 1 72
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[72,0,0,0,0,72,0,0,0,0,1,1,0,0,71,72],[72,0,0,0,71,1,0,0,0,0,61,67,0,0,12,12],[1,72,0,0,2,72,0,0,0,0,1,1,0,0,0,72],[1,72,0,0,0,72,0,0,0,0,1,1,0,0,0,72] >;

C3×Q8⋊D4 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes D_4
% in TeX

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

G:=SmallGroup(192,881);
// by ID

G=gap.SmallGroup(192,881);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,680,1094,4204,2111,172]);
// Polycyclic

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

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