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## G = C22×S3×D7order 336 = 24·3·7

### Direct product of C22, S3 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C22×S3×D7
 Chief series C1 — C7 — C21 — C3×D7 — S3×D7 — C2×S3×D7 — C22×S3×D7
 Lower central C21 — C22×S3×D7
 Upper central C1 — C22

Generators and relations for C22×S3×D7
G = < a,b,c,d,e,f | a2=b2=c3=d2=e7=f2=1, 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=e-1 >

Subgroups: 1564 in 268 conjugacy classes, 104 normal (14 characteristic)
C1, C2, C2, C3, C22, C22, S3, S3, C6, C6, C7, C23, D6, D6, C2×C6, C2×C6, D7, D7, C14, C14, C24, C21, C22×S3, C22×S3, C22×C6, D14, D14, C2×C14, C2×C14, S3×C7, C3×D7, D21, C42, S3×C23, C22×D7, C22×D7, C22×C14, S3×D7, C6×D7, S3×C14, D42, C2×C42, C23×D7, C2×S3×D7, C2×C6×D7, S3×C2×C14, C22×D21, C22×S3×D7
Quotients: C1, C2, C22, S3, C23, D6, D7, C24, C22×S3, D14, S3×C23, C22×D7, S3×D7, C23×D7, C2×S3×D7, C22×S3×D7

Smallest permutation representation of C22×S3×D7
On 84 points
Generators in S84
(1 48)(2 49)(3 43)(4 44)(5 45)(6 46)(7 47)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(43 50 57)(44 51 58)(45 52 59)(46 53 60)(47 54 61)(48 55 62)(49 56 63)(64 71 78)(65 72 79)(66 73 80)(67 74 81)(68 75 82)(69 76 83)(70 77 84)
(1 48)(2 49)(3 43)(4 44)(5 45)(6 46)(7 47)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)
(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 49)(7 48)(8 52)(9 51)(10 50)(11 56)(12 55)(13 54)(14 53)(15 59)(16 58)(17 57)(18 63)(19 62)(20 61)(21 60)(22 66)(23 65)(24 64)(25 70)(26 69)(27 68)(28 67)(29 73)(30 72)(31 71)(32 77)(33 76)(34 75)(35 74)(36 80)(37 79)(38 78)(39 84)(40 83)(41 82)(42 81)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 6A 6B 6C 6D 6E 6F 6G 7A 7B 7C 14A ··· 14I 14J ··· 14U 21A 21B 21C 42A ··· 42I order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 6 6 6 6 6 6 6 7 7 7 14 ··· 14 14 ··· 14 21 21 21 42 ··· 42 size 1 1 1 1 3 3 3 3 7 7 7 7 21 21 21 21 2 2 2 2 14 14 14 14 2 2 2 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 D7 D14 D14 S3×D7 C2×S3×D7 kernel C22×S3×D7 C2×S3×D7 C2×C6×D7 S3×C2×C14 C22×D21 C22×D7 D14 C2×C14 C22×S3 D6 C2×C6 C22 C2 # reps 1 12 1 1 1 1 6 1 3 18 3 3 9

Matrix representation of C22×S3×D7 in GL4(𝔽43) generated by

 1 0 0 0 0 1 0 0 0 0 42 0 0 0 0 42
,
 42 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42
,
 1 0 0 0 0 1 0 0 0 0 41 13 0 0 13 1
,
 1 0 0 0 0 1 0 0 0 0 42 0 0 0 13 1
,
 0 1 0 0 42 19 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 42 0 0 0 0 42
G:=sub<GL(4,GF(43))| [1,0,0,0,0,1,0,0,0,0,42,0,0,0,0,42],[42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,1,0,0,0,0,41,13,0,0,13,1],[1,0,0,0,0,1,0,0,0,0,42,13,0,0,0,1],[0,42,0,0,1,19,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,42,0,0,0,0,42] >;

C22×S3×D7 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times D_7
% in TeX

G:=Group("C2^2xS3xD7");
// GroupNames label

G:=SmallGroup(336,219);
// by ID

G=gap.SmallGroup(336,219);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,490,10373]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^2=e^7=f^2=1,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=e^-1>;
// generators/relations

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