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## G = D7×C3⋊D4order 336 = 24·3·7

### Direct product of D7 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D7×C3⋊D4
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C2×S3×D7 — D7×C3⋊D4
 Lower central C21 — C42 — D7×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for D7×C3⋊D4
G = < a,b,c,d,e | a7=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 684 in 108 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, C23, Dic3, Dic3, D6, D6, C2×C6, C2×C6, D7, D7, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C3⋊D4, C22×S3, C22×C6, Dic7, C28, D14, D14, C2×C14, C2×C14, S3×C7, C3×D7, C3×D7, D21, C42, C42, C2×C3⋊D4, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C22×D7, C7×Dic3, Dic21, S3×D7, C6×D7, C6×D7, S3×C14, D42, C2×C42, D4×D7, Dic3×D7, C21⋊D4, C3⋊D28, C7×C3⋊D4, C217D4, C2×S3×D7, C2×C6×D7, D7×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C3⋊D4, C22×S3, D14, C2×C3⋊D4, C22×D7, S3×D7, D4×D7, C2×S3×D7, D7×C3⋊D4

Smallest permutation representation of D7×C3⋊D4
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 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)
(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 27 69)(2 49 28 70)(3 43 22 64)(4 44 23 65)(5 45 24 66)(6 46 25 67)(7 47 26 68)(8 57 29 78)(9 58 30 79)(10 59 31 80)(11 60 32 81)(12 61 33 82)(13 62 34 83)(14 63 35 84)(15 50 36 71)(16 51 37 72)(17 52 38 73)(18 53 39 74)(19 54 40 75)(20 55 41 76)(21 56 42 77)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(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)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D7 C3⋊D4 D14 D14 D14 S3×D7 D4×D7 C2×S3×D7 D7×C3⋊D4 kernel D7×C3⋊D4 Dic3×D7 C21⋊D4 C3⋊D28 C7×C3⋊D4 C21⋊7D4 C2×S3×D7 C2×C6×D7 C22×D7 C3×D7 D14 C2×C14 C3⋊D4 D7 Dic3 D6 C2×C6 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 2 1 3 4 3 3 3 3 3 3 6

Matrix representation of D7×C3⋊D4 in GL6(𝔽337)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 193 1 0 0 0 0 335 110
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 0 0 0 34 303 0 0 0 0 143 303
,
 1 301 0 0 0 0 309 335 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 301 0 0 0 0 0 336 0 0 0 0 0 0 80 334 0 0 0 0 224 257 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 336 36 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 278 336 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,193,335,0,0,0,0,1,110],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,34,143,0,0,0,0,303,303],[1,309,0,0,0,0,301,335,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,301,336,0,0,0,0,0,0,80,224,0,0,0,0,334,257,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[336,0,0,0,0,0,36,1,0,0,0,0,0,0,1,278,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D7×C3⋊D4 in GAP, Magma, Sage, TeX

D_7\times C_3\rtimes D_4
% in TeX

G:=Group("D7xC3:D4");
// GroupNames label

G:=SmallGroup(336,161);
// by ID

G=gap.SmallGroup(336,161);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,116,490,10373]);
// Polycyclic

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

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