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## G = C7×A4⋊C4order 336 = 24·3·7

### Direct product of C7 and A4⋊C4

Aliases: C7×A4⋊C4, A4⋊C28, C14.6S4, (C7×A4)⋊3C4, (C2×A4).C14, C2.1(C7×S4), C23.(S3×C7), C22⋊(C7×Dic3), (A4×C14).3C2, (C2×C14)⋊1Dic3, (C22×C14).1S3, SmallGroup(336,117)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C7×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — A4×C14 — C7×A4⋊C4
 Lower central A4 — C7×A4⋊C4
 Upper central C1 — C14

Generators and relations for C7×A4⋊C4
G = < a,b,c,d,e | a7=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Smallest permutation representation of C7×A4⋊C4
On 84 points
Generators in S84
(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 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 22)(8 49)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 35)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(36 80)(37 81)(38 82)(39 83)(40 84)(41 78)(42 79)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 29)(7 30)(8 39)(9 40)(10 41)(11 42)(12 36)(13 37)(14 38)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(43 84)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 62)(51 63)(52 57)(53 58)(54 59)(55 60)(56 61)(64 76)(65 77)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 36 54)(2 37 55)(3 38 56)(4 39 50)(5 40 51)(6 41 52)(7 42 53)(8 70 26)(9 64 27)(10 65 28)(11 66 22)(12 67 23)(13 68 24)(14 69 25)(15 84 63)(16 78 57)(17 79 58)(18 80 59)(19 81 60)(20 82 61)(21 83 62)(29 44 77)(30 45 71)(31 46 72)(32 47 73)(33 48 74)(34 49 75)(35 43 76)
(1 18 31 23)(2 19 32 24)(3 20 33 25)(4 21 34 26)(5 15 35 27)(6 16 29 28)(7 17 30 22)(8 50 83 75)(9 51 84 76)(10 52 78 77)(11 53 79 71)(12 54 80 72)(13 55 81 73)(14 56 82 74)(36 59 46 67)(37 60 47 68)(38 61 48 69)(39 62 49 70)(40 63 43 64)(41 57 44 65)(42 58 45 66)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 7A ··· 7F 14A ··· 14F 14G ··· 14R 21A ··· 21F 28A ··· 28X 42A ··· 42F order 1 2 2 2 3 4 4 4 4 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 size 1 1 3 3 8 6 6 6 6 8 1 ··· 1 1 ··· 1 3 ··· 3 8 ··· 8 6 ··· 6 8 ··· 8

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + - + image C1 C2 C4 C7 C14 C28 S3 Dic3 S3×C7 C7×Dic3 S4 A4⋊C4 C7×S4 C7×A4⋊C4 kernel C7×A4⋊C4 A4×C14 C7×A4 A4⋊C4 C2×A4 A4 C22×C14 C2×C14 C23 C22 C14 C7 C2 C1 # reps 1 1 2 6 6 12 1 1 6 6 2 2 12 12

Matrix representation of C7×A4⋊C4 in GL5(𝔽337)

 1 0 0 0 0 0 1 0 0 0 0 0 295 0 0 0 0 0 295 0 0 0 0 0 295
,
 1 0 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 336
,
 336 336 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 189 0 0 0 0 148 148 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 336 0

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,1,0,0,0,0,0,295,0,0,0,0,0,295,0,0,0,0,0,295],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,336],[336,1,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[189,148,0,0,0,0,148,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,336,0] >;

C7×A4⋊C4 in GAP, Magma, Sage, TeX

C_7\times A_4\rtimes C_4
% in TeX

G:=Group("C7xA4:C4");
// GroupNames label

G:=SmallGroup(336,117);
// by ID

G=gap.SmallGroup(336,117);
# by ID

G:=PCGroup([6,-2,-7,-2,-3,-2,2,84,1347,5044,202,3029,347]);
// Polycyclic

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

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