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## G = C4×C7⋊A4order 336 = 24·3·7

### Direct product of C4 and C7⋊A4

Aliases: C4×C7⋊A4, C28⋊A4, C72(C4×A4), (C2×C14)⋊7C12, C14.4(C2×A4), (C22×C28)⋊3C3, (C22×C14).8C6, C22⋊(C4×C7⋊C3), C2.1(C2×C7⋊A4), (C22×C4)⋊(C7⋊C3), C23.(C2×C7⋊C3), (C2×C7⋊A4).2C2, SmallGroup(336,171)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C7⋊A4
G = < a,b,c,d,e | a4=b7=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, ece-1=cd=dc, ede-1=c >

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 7A 7B 12A 12B 12C 12D 14A ··· 14N 28A ··· 28P order 1 2 2 2 3 3 4 4 4 4 6 6 7 7 12 12 12 12 14 ··· 14 28 ··· 28 size 1 1 3 3 28 28 1 1 3 3 28 28 3 3 28 28 28 28 3 ··· 3 3 ··· 3

48 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C4 C6 C12 A4 C7⋊C3 C2×A4 C2×C7⋊C3 C4×A4 C7⋊A4 C4×C7⋊C3 C2×C7⋊A4 C4×C7⋊A4 kernel C4×C7⋊A4 C2×C7⋊A4 C22×C28 C7⋊A4 C22×C14 C2×C14 C28 C22×C4 C14 C23 C7 C4 C22 C2 C1 # reps 1 1 2 2 2 4 1 2 1 2 2 6 4 6 12

Matrix representation of C4×C7⋊A4 in GL3(𝔽337) generated by

 189 0 0 0 189 0 0 0 189
,
 8 0 0 0 64 0 0 0 52
,
 336 0 0 0 336 0 0 0 1
,
 336 0 0 0 1 0 0 0 336
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(337))| [189,0,0,0,189,0,0,0,189],[8,0,0,0,64,0,0,0,52],[336,0,0,0,336,0,0,0,1],[336,0,0,0,1,0,0,0,336],[0,0,1,1,0,0,0,1,0] >;

C4×C7⋊A4 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes A_4
% in TeX

G:=Group("C4xC7:A4");
// GroupNames label

G:=SmallGroup(336,171);
// by ID

G=gap.SmallGroup(336,171);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-7,36,297,550,1739]);
// Polycyclic

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

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