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G = (C22×S3)⋊Q8order 192 = 26·3

1st semidirect product of C22×S3 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C22×S3)⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C22×S3)⋊Q8
 Lower central C3 — C22×C6 — (C22×S3)⋊Q8
 Upper central C1 — C23 — C2.C42

Generators and relations for (C22×S3)⋊Q8
G = < a,b,c,d,e,f | a2=b2=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd=ece-1=fcf-1=c-1, ede-1=abd, fef-1=e-1 >

Subgroups: 624 in 202 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C22×Q8, D6⋊C4, C2×Dic6, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23⋊Q8, C6.C42, C3×C2.C42, C2×D6⋊C4, C2×D6⋊C4, C22×Dic6, (C22×S3)⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22≀C2, C22⋊Q8, C4.4D4, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, C23⋊Q8, C427S3, D6⋊D4, C23.11D6, D6⋊Q8, C4.D12, (C22×S3)⋊Q8

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

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

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

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

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 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 S3 D4 D4 Q8 D6 C4○D4 D12 C4○D12 S3×D4 D4⋊2S3 S3×Q8 kernel (C22×S3)⋊Q8 C6.C42 C3×C2.C42 C2×D6⋊C4 C22×Dic6 C2.C42 C2×Dic3 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 2 1 3 1 1 4 2 2 3 6 4 8 2 1 1

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 4 11 0 0 0 0 2 9 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;

(C22×S3)⋊Q8 in GAP, Magma, Sage, TeX

(C_2^2\times S_3)\rtimes Q_8
% in TeX

G:=Group("(C2^2xS3):Q8");
// GroupNames label

G:=SmallGroup(192,232);
// by ID

G=gap.SmallGroup(192,232);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,64,926,387,268,6278]);
// Polycyclic

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

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