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## G = C3×C7⋊D4order 168 = 23·3·7

### Direct product of C3 and C7⋊D4

Aliases: C3×C7⋊D4, C218D4, D145C6, Dic74C6, C6.17D14, C42.17C22, C75(C3×D4), (C2×C6)⋊1D7, (C2×C42)⋊4C2, (C6×D7)⋊5C2, C2.5(C6×D7), (C2×C14)⋊10C6, C222(C3×D7), C14.13(C2×C6), (C3×Dic7)⋊4C2, SmallGroup(168,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C3×C7⋊D4
 Chief series C1 — C7 — C14 — C42 — C6×D7 — C3×C7⋊D4
 Lower central C7 — C14 — C3×C7⋊D4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C7⋊D4
G = < a,b,c,d | a3=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C3×C7⋊D4
On 84 points
Generators in S84
(1 29 15)(2 30 16)(3 31 17)(4 32 18)(5 33 19)(6 34 20)(7 35 21)(8 36 22)(9 37 23)(10 38 24)(11 39 25)(12 40 26)(13 41 27)(14 42 28)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)
(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 43 8 50)(2 49 9 56)(3 48 10 55)(4 47 11 54)(5 46 12 53)(6 45 13 52)(7 44 14 51)(15 57 22 64)(16 63 23 70)(17 62 24 69)(18 61 25 68)(19 60 26 67)(20 59 27 66)(21 58 28 65)(29 71 36 78)(30 77 37 84)(31 76 38 83)(32 75 39 82)(33 74 40 81)(34 73 41 80)(35 72 42 79)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 50)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 64)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 78)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)

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

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

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

C3×C7⋊D4 is a maximal subgroup of   Dic7.D6  C42.C23  D6⋊D14  C3×D4×D7

51 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7A 7B 7C 12A 12B 14A ··· 14I 21A ··· 21F 42A ··· 42R order 1 2 2 2 3 3 4 6 6 6 6 6 6 7 7 7 12 12 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 2 14 1 1 14 1 1 2 2 14 14 2 2 2 14 14 2 ··· 2 2 ··· 2 2 ··· 2

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D7 C3×D4 D14 C3×D7 C7⋊D4 C6×D7 C3×C7⋊D4 kernel C3×C7⋊D4 C3×Dic7 C6×D7 C2×C42 C7⋊D4 Dic7 D14 C2×C14 C21 C2×C6 C7 C6 C22 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 3 2 3 6 6 6 12

Matrix representation of C3×C7⋊D4 in GL2(𝔽43) generated by

 36 0 0 36
,
 18 10 16 40
,
 0 7 6 0
,
 11 21 25 32
G:=sub<GL(2,GF(43))| [36,0,0,36],[18,16,10,40],[0,6,7,0],[11,25,21,32] >;

C3×C7⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes D_4
% in TeX

G:=Group("C3xC7:D4");
// GroupNames label

G:=SmallGroup(168,28);
// by ID

G=gap.SmallGroup(168,28);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,141,3604]);
// Polycyclic

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

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