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## G = Q8×C3⋊D4order 192 = 26·3

### Direct product of Q8 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Q8×C3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — Q8×C3⋊D4
 Lower central C3 — C2×C6 — Q8×C3⋊D4
 Upper central C1 — C22 — C22×Q8

Generators and relations for Q8×C3⋊D4
G = < a,b,c,d,e | a4=c3=d4=e2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 600 in 280 conjugacy classes, 123 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, S3×Q8, C2×C3⋊D4, C22×C12, C6×Q8, C6×Q8, C6×Q8, D4×Q8, C12.48D4, C4×C3⋊D4, Dic3⋊Q8, Q8×Dic3, D63Q8, C2×S3×Q8, Q8×C2×C6, Q8×C3⋊D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, C3⋊D4, C22×S3, C22×D4, C22×Q8, 2- 1+4, S3×Q8, C2×C3⋊D4, S3×C23, D4×Q8, C2×S3×Q8, Q8.15D6, C22×C3⋊D4, Q8×C3⋊D4

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 C3⋊D4 2- 1+4 S3×Q8 Q8.15D6 kernel Q8×C3⋊D4 C12.48D4 C4×C3⋊D4 Dic3⋊Q8 Q8×Dic3 D6⋊3Q8 C2×S3×Q8 Q8×C2×C6 C22×Q8 C3⋊D4 C3×Q8 C22×C4 C2×Q8 Q8 C6 C22 C2 # reps 1 3 3 3 1 3 1 1 1 4 4 3 4 8 1 2 2

Matrix representation of Q8×C3⋊D4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 12 3 0 0 8 1
,
 12 0 0 0 0 12 0 0 0 0 1 1 0 0 11 12
,
 0 12 0 0 1 12 0 0 0 0 1 0 0 0 0 1
,
 4 11 0 0 2 9 0 0 0 0 12 0 0 0 0 12
,
 0 12 0 0 12 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,8,0,0,3,1],[12,0,0,0,0,12,0,0,0,0,1,11,0,0,1,12],[0,1,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[4,2,0,0,11,9,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,12,0,0,0,0,0,1,0,0,0,0,1] >;

Q8×C3⋊D4 in GAP, Magma, Sage, TeX

Q_8\times C_3\rtimes D_4
% in TeX

G:=Group("Q8xC3:D4");
// GroupNames label

G:=SmallGroup(192,1374);
// by ID

G=gap.SmallGroup(192,1374);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,6278]);
// Polycyclic

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

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