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## G = C4×S3×D7order 336 = 24·3·7

### Direct product of C4, S3 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C4×S3×D7
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C2×S3×D7 — C4×S3×D7
 Lower central C21 — C4×S3×D7
 Upper central C1 — C4

Generators and relations for C4×S3×D7
G = < a,b,c,d,e | a4=b3=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 580 in 108 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C7, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, D7, D7, C14, C14, C22×C4, C21, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, Dic7, Dic7, C28, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, S3×C2×C4, C4×D7, C4×D7, C2×Dic7, C2×C28, C22×D7, C7×Dic3, C3×Dic7, Dic21, C84, S3×D7, C6×D7, S3×C14, D42, C2×C4×D7, Dic3×D7, S3×Dic7, D21⋊C4, C12×D7, S3×C28, C4×D21, C2×S3×D7, C4×S3×D7
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, D7, C22×C4, C4×S3, C22×S3, D14, S3×C2×C4, C4×D7, C22×D7, S3×D7, C2×C4×D7, C2×S3×D7, C4×S3×D7

Smallest permutation representation of C4×S3×D7
On 84 points
Generators in S84
(1 69 27 48)(2 70 28 49)(3 64 22 43)(4 65 23 44)(5 66 24 45)(6 67 25 46)(7 68 26 47)(8 71 29 50)(9 72 30 51)(10 73 31 52)(11 74 32 53)(12 75 33 54)(13 76 34 55)(14 77 35 56)(15 78 36 57)(16 79 37 58)(17 80 38 59)(18 81 39 60)(19 82 40 61)(20 83 41 62)(21 84 42 63)
(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 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 31)(9 30)(10 29)(11 35)(12 34)(13 33)(14 32)(15 38)(16 37)(17 36)(18 42)(19 41)(20 40)(21 39)(43 66)(44 65)(45 64)(46 70)(47 69)(48 68)(49 67)(50 73)(51 72)(52 71)(53 77)(54 76)(55 75)(56 74)(57 80)(58 79)(59 78)(60 84)(61 83)(62 82)(63 81)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A 14B 14C 14D ··· 14I 21A 21B 21C 28A ··· 28F 28G ··· 28L 42A 42B 42C 84A ··· 84F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 7 7 7 12 12 12 12 14 14 14 14 ··· 14 21 21 21 28 ··· 28 28 ··· 28 42 42 42 84 ··· 84 size 1 1 3 3 7 7 21 21 2 1 1 3 3 7 7 21 21 2 14 14 2 2 2 2 2 14 14 2 2 2 6 ··· 6 4 4 4 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 D7 C4×S3 D14 D14 D14 C4×D7 S3×D7 C2×S3×D7 C4×S3×D7 kernel C4×S3×D7 Dic3×D7 S3×Dic7 D21⋊C4 C12×D7 S3×C28 C4×D21 C2×S3×D7 S3×D7 C4×D7 Dic7 C28 D14 C4×S3 D7 Dic3 C12 D6 S3 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 8 1 1 1 1 3 4 3 3 3 12 3 3 6

Matrix representation of C4×S3×D7 in GL4(𝔽337) generated by

 189 0 0 0 0 189 0 0 0 0 189 0 0 0 0 189
,
 335 28 0 0 36 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 301 336 0 0 0 0 336 0 0 0 0 336
,
 1 0 0 0 0 1 0 0 0 0 336 305 0 0 336 304
,
 336 0 0 0 0 336 0 0 0 0 110 260 0 0 144 227
G:=sub<GL(4,GF(337))| [189,0,0,0,0,189,0,0,0,0,189,0,0,0,0,189],[335,36,0,0,28,1,0,0,0,0,1,0,0,0,0,1],[1,301,0,0,0,336,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,336,336,0,0,305,304],[336,0,0,0,0,336,0,0,0,0,110,144,0,0,260,227] >;

C4×S3×D7 in GAP, Magma, Sage, TeX

C_4\times S_3\times D_7
% in TeX

G:=Group("C4xS3xD7");
// GroupNames label

G:=SmallGroup(336,147);
// by ID

G=gap.SmallGroup(336,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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