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## G = C22×S3×Q8order 192 = 26·3

### Direct product of C22, S3 and Q8

Series: Derived Chief Lower central Upper central

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

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, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1544 in 850 conjugacy classes, 503 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, Q8, C23, C23, Dic3, C12, D6, C2×C6, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22×C6, C23×C4, C22×Q8, C22×Q8, C2×Dic6, S3×C2×C4, S3×Q8, C22×Dic3, C22×C12, C6×Q8, S3×C23, Q8×C23, C22×Dic6, S3×C22×C4, C2×S3×Q8, Q8×C2×C6, C22×S3×Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, C25, S3×Q8, S3×C23, Q8×C23, C2×S3×Q8, S3×C24, C22×S3×Q8

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4L 4M ··· 4X 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 2 2 ··· 2 6 ··· 6 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 Q8 D6 D6 S3×Q8 kernel C22×S3×Q8 C22×Dic6 S3×C22×C4 C2×S3×Q8 Q8×C2×C6 C22×Q8 C22×S3 C22×C4 C2×Q8 C22 # reps 1 3 3 24 1 1 8 3 12 4

Matrix representation of C22×S3×Q8 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
,
 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 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 12 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 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 5 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 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0

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],[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],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C22×S3×Q8 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times Q_8
% in TeX

G:=Group("C2^2xS3xQ8");
// GroupNames label

G:=SmallGroup(192,1517);
// by ID

G=gap.SmallGroup(192,1517);
# by ID

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

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