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## G = (Q8×Dic3)⋊C2order 192 = 26·3

### 10th semidirect product of Q8×Dic3 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (Q8×Dic3)⋊C2
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — (Q8×Dic3)⋊C2
 Lower central C3 — C2×C6 — (Q8×Dic3)⋊C2
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for (Q8×Dic3)⋊C2
G = < a,b,c,d,e | a4=c6=e2=1, b2=a2, d2=c3, bab-1=a-1, ac=ca, ad=da, eae=ac3, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, de=ed >

Subgroups: 448 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×C6, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×Q8, C23.37C23, C23.16D6, Dic3.D4, Dic6⋊C4, Dic6⋊C4, C12⋊Q8, Dic3.Q8, C2×C4×Dic3, C12.48D4, Dic3⋊Q8, Q8×Dic3, C3×C22⋊Q8, (Q8×Dic3)⋊C2
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, D42S3, S3×Q8, S3×C23, C23.37C23, C2×D42S3, C2×S3×Q8, S3×C4○D4, (Q8×Dic3)⋊C2

Smallest permutation representation of (Q8×Dic3)⋊C2
On 96 points
Generators in S96
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 37 31 43)(26 38 32 44)(27 39 33 45)(28 40 34 46)(29 41 35 47)(30 42 36 48)(49 67 55 61)(50 68 56 62)(51 69 57 63)(52 70 58 64)(53 71 59 65)(54 72 60 66)(73 85 79 91)(74 86 80 92)(75 87 81 93)(76 88 82 94)(77 89 83 95)(78 90 84 96)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)(49 79 55 73)(50 80 56 74)(51 81 57 75)(52 82 58 76)(53 83 59 77)(54 84 60 78)(61 91 67 85)(62 92 68 86)(63 93 69 87)(64 94 70 88)(65 95 71 89)(66 96 72 90)
(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 50 4 53)(2 49 5 52)(3 54 6 51)(7 56 10 59)(8 55 11 58)(9 60 12 57)(13 62 16 65)(14 61 17 64)(15 66 18 63)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 74 28 77)(26 73 29 76)(27 78 30 75)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(61 64)(62 65)(63 66)(67 70)(68 71)(69 72)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 D6 C4○D4 C4○D4 D4⋊2S3 S3×Q8 S3×C4○D4 kernel (Q8×Dic3)⋊C2 C23.16D6 Dic3.D4 Dic6⋊C4 C12⋊Q8 Dic3.Q8 C2×C4×Dic3 C12.48D4 Dic3⋊Q8 Q8×Dic3 C3×C22⋊Q8 C22⋊Q8 C2×Dic3 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 Dic3 C12 C4 C22 C2 # reps 1 2 2 3 1 2 1 1 1 1 1 1 4 2 3 1 1 4 4 2 2 2

Matrix representation of (Q8×Dic3)⋊C2 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;

(Q8×Dic3)⋊C2 in GAP, Magma, Sage, TeX

(Q_8\times {\rm Dic}_3)\rtimes C_2
% in TeX

G:=Group("(Q8xDic3):C2");
// GroupNames label

G:=SmallGroup(192,1181);
// by ID

G=gap.SmallGroup(192,1181);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,570,185,80,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^6=e^2=1,b^2=a^2,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e=a*c^3,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,d*e=e*d>;
// generators/relations

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