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## G = C22×Q8⋊3S3order 192 = 26·3

### Direct product of C22 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×Q8⋊3S3
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C22×C4 — C22×Q8⋊3S3
 Lower central C3 — C6 — C22×Q8⋊3S3
 Upper central C1 — C23 — C22×Q8

Generators and relations for C22×Q83S3
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, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 1864 in 890 conjugacy classes, 463 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C23×C4, C22×D4, C22×Q8, C2×C4○D4, S3×C2×C4, C2×D12, Q83S3, C22×Dic3, C22×C12, C6×Q8, S3×C23, C22×C4○D4, S3×C22×C4, C22×D12, C2×Q83S3, Q8×C2×C6, C22×Q83S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C25, Q83S3, S3×C23, C22×C4○D4, C2×Q83S3, S3×C24, C22×Q83S3

Smallest permutation representation of C22×Q83S3
On 96 points
Generators in S96
(1 60)(2 57)(3 58)(4 59)(5 51)(6 52)(7 49)(8 50)(9 55)(10 56)(11 53)(12 54)(13 63)(14 64)(15 61)(16 62)(17 67)(18 68)(19 65)(20 66)(21 71)(22 72)(23 69)(24 70)(25 75)(26 76)(27 73)(28 74)(29 79)(30 80)(31 77)(32 78)(33 83)(34 84)(35 81)(36 82)(37 87)(38 88)(39 85)(40 86)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)
(1 36)(2 33)(3 34)(4 35)(5 75)(6 76)(7 73)(8 74)(9 79)(10 80)(11 77)(12 78)(13 39)(14 40)(15 37)(16 38)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)(25 51)(26 52)(27 49)(28 50)(29 55)(30 56)(31 53)(32 54)(57 83)(58 84)(59 81)(60 82)(61 87)(62 88)(63 85)(64 86)(65 91)(66 92)(67 89)(68 90)(69 95)(70 96)(71 93)(72 94)
(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 72 3 70)(2 71 4 69)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 78 15 80)(14 77 16 79)(17 74 19 76)(18 73 20 75)(21 59 23 57)(22 58 24 60)(25 68 27 66)(26 67 28 65)(29 64 31 62)(30 63 32 61)(33 93 35 95)(34 96 36 94)(45 83 47 81)(46 82 48 84)(49 92 51 90)(50 91 52 89)(53 88 55 86)(54 87 56 85)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 93 10)(6 94 11)(7 95 12)(8 96 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 54 49)(46 55 50)(47 56 51)(48 53 52)(57 61 66)(58 62 67)(59 63 68)(60 64 65)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 85 90)(82 86 91)(83 87 92)(84 88 89)
(1 58)(2 57)(3 60)(4 59)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 71)(22 70)(23 69)(24 72)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 83)(34 82)(35 81)(36 84)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 95)(46 94)(47 93)(48 96)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 3 4A ··· 4L 4M ··· 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 ··· 6 2 2 ··· 2 3 ··· 3 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 C4○D4 Q8⋊3S3 kernel C22×Q8⋊3S3 S3×C22×C4 C22×D12 C2×Q8⋊3S3 Q8×C2×C6 C22×Q8 C22×C4 C2×Q8 C2×C6 C22 # reps 1 3 3 24 1 1 3 12 8 4

Matrix representation of C22×Q83S3 in GL6(𝔽13)

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

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

C22×Q83S3 in GAP, Magma, Sage, TeX

C_2^2\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C2^2xQ8:3S3");
// GroupNames label

G:=SmallGroup(192,1518);
// by ID

G=gap.SmallGroup(192,1518);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,1684,235,102,6278]);
// 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,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*c*f=c^-1,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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