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## G = (C22×Q8)⋊9S3order 192 = 26·3

### 2nd semidirect product of C22×Q8 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C22×Q8)⋊9S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C22×Q8)⋊9S3
 Lower central C3 — C22×C6 — (C22×Q8)⋊9S3
 Upper central C1 — C23 — C22×Q8

Generators and relations for (C22×Q8)⋊9S3
G = < a,b,c,d,e,f | a2=b2=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fdf=bd=db, be=eb, bf=fb, dcd-1=c-1, ce=ec, fcf=abc-1, de=ed, fef=e-1 >

Subgroups: 568 in 202 conjugacy classes, 65 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×2], C3, C4 [×9], C22, C22 [×6], C22 [×10], S3 [×2], C6, C6 [×6], C2×C4 [×6], C2×C4 [×15], Q8 [×8], C23, C23 [×8], Dic3 [×3], C12 [×6], D6 [×10], C2×C6, C2×C6 [×6], C22⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C2×Q8 [×6], C24, C2×Dic3 [×9], C2×C12 [×6], C2×C12 [×6], C3×Q8 [×8], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42 [×3], C2×C22⋊C4 [×3], C22×Q8, D6⋊C4 [×6], C22×Dic3 [×3], C22×C12 [×3], C6×Q8 [×6], S3×C23, C23⋊Q8, C6.C42 [×3], C2×D6⋊C4 [×3], Q8×C2×C6, (C22×Q8)⋊9S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], S3×Q8, Q83S3 [×3], C2×C3⋊D4 [×3], C23⋊Q8, D63Q8 [×3], C12.23D4 [×3], C244S3, (C22×Q8)⋊9S3

Smallest permutation representation of (C22×Q8)⋊9S3
On 96 points
Generators in S96
(1 23)(2 24)(3 21)(4 22)(5 25)(6 26)(7 27)(8 28)(9 35)(10 36)(11 33)(12 34)(13 41)(14 42)(15 43)(16 44)(17 46)(18 47)(19 48)(20 45)(29 62)(30 63)(31 64)(32 61)(37 70)(38 71)(39 72)(40 69)(49 96)(50 93)(51 94)(52 95)(53 77)(54 78)(55 79)(56 80)(57 83)(58 84)(59 81)(60 82)(65 85)(66 86)(67 87)(68 88)(73 92)(74 89)(75 90)(76 91)
(1 11)(2 12)(3 9)(4 10)(5 18)(6 19)(7 20)(8 17)(13 64)(14 61)(15 62)(16 63)(21 35)(22 36)(23 33)(24 34)(25 47)(26 48)(27 45)(28 46)(29 43)(30 44)(31 41)(32 42)(37 76)(38 73)(39 74)(40 75)(49 54)(50 55)(51 56)(52 53)(57 66)(58 67)(59 68)(60 65)(69 90)(70 91)(71 92)(72 89)(77 95)(78 96)(79 93)(80 94)(81 88)(82 85)(83 86)(84 87)
(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 87 3 85)(2 86 4 88)(5 93 7 95)(6 96 8 94)(9 82 11 84)(10 81 12 83)(13 70 15 72)(14 69 16 71)(17 80 19 78)(18 79 20 77)(21 65 23 67)(22 68 24 66)(25 50 27 52)(26 49 28 51)(29 74 31 76)(30 73 32 75)(33 58 35 60)(34 57 36 59)(37 43 39 41)(38 42 40 44)(45 53 47 55)(46 56 48 54)(61 90 63 92)(62 89 64 91)
(1 27 13)(2 28 14)(3 25 15)(4 26 16)(5 43 21)(6 44 22)(7 41 23)(8 42 24)(9 47 62)(10 48 63)(11 45 64)(12 46 61)(17 32 34)(18 29 35)(19 30 36)(20 31 33)(37 67 95)(38 68 96)(39 65 93)(40 66 94)(49 71 88)(50 72 85)(51 69 86)(52 70 87)(53 91 84)(54 92 81)(55 89 82)(56 90 83)(57 80 75)(58 77 76)(59 78 73)(60 79 74)
(2 36)(4 34)(5 43)(6 61)(7 41)(8 63)(10 24)(12 22)(13 27)(14 19)(15 25)(16 17)(18 29)(20 31)(26 32)(28 30)(37 77)(38 51)(39 79)(40 49)(42 48)(44 46)(45 64)(47 62)(50 89)(52 91)(53 70)(54 75)(55 72)(56 73)(57 81)(58 67)(59 83)(60 65)(66 88)(68 86)(69 96)(71 94)(74 93)(76 95)(78 90)(80 92)(82 85)(84 87)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A ··· 4F 4G ··· 4L 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - + - + image C1 C2 C2 C2 S3 D4 Q8 D6 C4○D4 C3⋊D4 S3×Q8 Q8⋊3S3 kernel (C22×Q8)⋊9S3 C6.C42 C2×D6⋊C4 Q8×C2×C6 C22×Q8 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 3 3 1 1 6 2 3 6 12 1 3

Matrix representation of (C22×Q8)⋊9S3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 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 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 0 0 0 0 0 12
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 7 10 0 0 0 0 3 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 6 3 0 0 0 0 10 7 0 0 0 0 0 0 11 9 0 0 0 0 4 2
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 9 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,7,3,0,0,0,0,10,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,9,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

(C22×Q8)⋊9S3 in GAP, Magma, Sage, TeX

(C_2^2\times Q_8)\rtimes_9S_3
% in TeX

G:=Group("(C2^2xQ8):9S3");
// GroupNames label

G:=SmallGroup(192,790);
// by ID

G=gap.SmallGroup(192,790);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,254,387,184,6278]);
// Polycyclic

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

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