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## G = D7×D12order 336 = 24·3·7

### Direct product of D7 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D7×D12
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C2×S3×D7 — D7×D12
 Lower central C21 — C42 — D7×D12
 Upper central C1 — C2 — C4

Generators and relations for D7×D12
G = < a,b,c,d | a7=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 788 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, C23, C12, C12, D6, D6, C2×C6, D7, D7, C14, C14, C2×D4, C21, D12, D12, C2×C12, C22×S3, Dic7, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, C2×D12, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C3×Dic7, C84, S3×D7, C6×D7, S3×C14, D42, D4×D7, C7⋊D12, C12×D7, C7×D12, D84, C2×S3×D7, D7×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, D12, C22×S3, D14, C2×D12, C22×D7, S3×D7, D4×D7, C2×S3×D7, D7×D12

Smallest permutation representation of D7×D12
On 84 points
Generators in S84
(1 53 68 77 18 37 30)(2 54 69 78 19 38 31)(3 55 70 79 20 39 32)(4 56 71 80 21 40 33)(5 57 72 81 22 41 34)(6 58 61 82 23 42 35)(7 59 62 83 24 43 36)(8 60 63 84 13 44 25)(9 49 64 73 14 45 26)(10 50 65 74 15 46 27)(11 51 66 75 16 47 28)(12 52 67 76 17 48 29)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 69)(14 70)(15 71)(16 72)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(37 59)(38 60)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(73 79)(74 80)(75 81)(76 82)(77 83)(78 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 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 S3 D4 D6 D6 D6 D7 D12 D14 D14 S3×D7 D4×D7 C2×S3×D7 D7×D12 kernel D7×D12 C7⋊D12 C12×D7 C7×D12 D84 C2×S3×D7 C4×D7 C3×D7 Dic7 C28 D14 D12 D7 C12 D6 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 2 1 1 1 3 4 3 6 3 3 3 6

Matrix representation of D7×D12 in GL4(𝔽337) generated by

 109 1 0 0 251 194 0 0 0 0 1 0 0 0 0 1
,
 194 303 0 0 86 143 0 0 0 0 336 0 0 0 0 336
,
 1 0 0 0 0 1 0 0 0 0 292 244 0 0 29 0
,
 1 0 0 0 0 1 0 0 0 0 0 244 0 0 308 0
G:=sub<GL(4,GF(337))| [109,251,0,0,1,194,0,0,0,0,1,0,0,0,0,1],[194,86,0,0,303,143,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,292,29,0,0,244,0],[1,0,0,0,0,1,0,0,0,0,0,308,0,0,244,0] >;

D7×D12 in GAP, Magma, Sage, TeX

D_7\times D_{12}
% in TeX

G:=Group("D7xD12");
// GroupNames label

G:=SmallGroup(336,148);
// by ID

G=gap.SmallGroup(336,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,116,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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