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## G = D7×C4⋊D4order 448 = 26·7

### Direct product of D7 and C4⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7×C4⋊D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — D7×C4⋊D4
 Lower central C7 — C2×C14 — D7×C4⋊D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for D7×C4⋊D4
G = < a,b,c,d,e | a7=b2=c4=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: 2380 in 426 conjugacy classes, 121 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D7, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C23×C4, C22×D4, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, C2×C4⋊D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, C23×D7, D7×C22⋊C4, D14⋊D4, D7×C4⋊C4, C4⋊D28, C287D4, C282D4, Dic7⋊D4, C7×C4⋊D4, D7×C22×C4, C2×D4×D7, C2×D4×D7, D7×C4⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C4⋊D4, C22×D4, C2×C4○D4, C22×D7, C2×C4⋊D4, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D7×C4⋊D4

Smallest permutation representation of D7×C4⋊D4
On 112 points
Generators in S112
(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)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)
(1 76 20 62)(2 77 21 63)(3 71 15 57)(4 72 16 58)(5 73 17 59)(6 74 18 60)(7 75 19 61)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 92 43 106)(30 93 44 107)(31 94 45 108)(32 95 46 109)(33 96 47 110)(34 97 48 111)(35 98 49 112)(36 85 50 99)(37 86 51 100)(38 87 52 101)(39 88 53 102)(40 89 54 103)(41 90 55 104)(42 91 56 105)
(1 90 27 111)(2 91 28 112)(3 85 22 106)(4 86 23 107)(5 87 24 108)(6 88 25 109)(7 89 26 110)(8 92 15 99)(9 93 16 100)(10 94 17 101)(11 95 18 102)(12 96 19 103)(13 97 20 104)(14 98 21 105)(29 57 50 78)(30 58 51 79)(31 59 52 80)(32 60 53 81)(33 61 54 82)(34 62 55 83)(35 63 56 84)(36 64 43 71)(37 65 44 72)(38 66 45 73)(39 67 46 74)(40 68 47 75)(41 69 48 76)(42 70 49 77)
(1 97)(2 98)(3 92)(4 93)(5 94)(6 95)(7 96)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)

G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109), (1,76,20,62)(2,77,21,63)(3,71,15,57)(4,72,16,58)(5,73,17,59)(6,74,18,60)(7,75,19,61)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,92,43,106)(30,93,44,107)(31,94,45,108)(32,95,46,109)(33,96,47,110)(34,97,48,111)(35,98,49,112)(36,85,50,99)(37,86,51,100)(38,87,52,101)(39,88,53,102)(40,89,54,103)(41,90,55,104)(42,91,56,105), (1,90,27,111)(2,91,28,112)(3,85,22,106)(4,86,23,107)(5,87,24,108)(6,88,25,109)(7,89,26,110)(8,92,15,99)(9,93,16,100)(10,94,17,101)(11,95,18,102)(12,96,19,103)(13,97,20,104)(14,98,21,105)(29,57,50,78)(30,58,51,79)(31,59,52,80)(32,60,53,81)(33,61,54,82)(34,62,55,83)(35,63,56,84)(36,64,43,71)(37,65,44,72)(38,66,45,73)(39,67,46,74)(40,68,47,75)(41,69,48,76)(42,70,49,77), (1,97)(2,98)(3,92)(4,93)(5,94)(6,95)(7,96)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)>;

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)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109), (1,76,20,62)(2,77,21,63)(3,71,15,57)(4,72,16,58)(5,73,17,59)(6,74,18,60)(7,75,19,61)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,92,43,106)(30,93,44,107)(31,94,45,108)(32,95,46,109)(33,96,47,110)(34,97,48,111)(35,98,49,112)(36,85,50,99)(37,86,51,100)(38,87,52,101)(39,88,53,102)(40,89,54,103)(41,90,55,104)(42,91,56,105), (1,90,27,111)(2,91,28,112)(3,85,22,106)(4,86,23,107)(5,87,24,108)(6,88,25,109)(7,89,26,110)(8,92,15,99)(9,93,16,100)(10,94,17,101)(11,95,18,102)(12,96,19,103)(13,97,20,104)(14,98,21,105)(29,57,50,78)(30,58,51,79)(31,59,52,80)(32,60,53,81)(33,61,54,82)(34,62,55,83)(35,63,56,84)(36,64,43,71)(37,65,44,72)(38,66,45,73)(39,67,46,74)(40,68,47,75)(41,69,48,76)(42,70,49,77), (1,97)(2,98)(3,92)(4,93)(5,94)(6,95)(7,96)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70) );

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),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109)], [(1,76,20,62),(2,77,21,63),(3,71,15,57),(4,72,16,58),(5,73,17,59),(6,74,18,60),(7,75,19,61),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,92,43,106),(30,93,44,107),(31,94,45,108),(32,95,46,109),(33,96,47,110),(34,97,48,111),(35,98,49,112),(36,85,50,99),(37,86,51,100),(38,87,52,101),(39,88,53,102),(40,89,54,103),(41,90,55,104),(42,91,56,105)], [(1,90,27,111),(2,91,28,112),(3,85,22,106),(4,86,23,107),(5,87,24,108),(6,88,25,109),(7,89,26,110),(8,92,15,99),(9,93,16,100),(10,94,17,101),(11,95,18,102),(12,96,19,103),(13,97,20,104),(14,98,21,105),(29,57,50,78),(30,58,51,79),(31,59,52,80),(32,60,53,81),(33,61,54,82),(34,62,55,83),(35,63,56,84),(36,64,43,71),(37,65,44,72),(38,66,45,73),(39,67,46,74),(40,68,47,75),(41,69,48,76),(42,70,49,77)], [(1,97),(2,98),(3,92),(4,93),(5,94),(6,95),(7,96),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70)]])

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14I 14J ··· 14O 14P ··· 14U 28A ··· 28L 28M ··· 28R order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 4 7 7 7 7 14 14 28 28 2 2 2 2 4 4 14 14 14 14 28 28 2 2 2 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4 8 ··· 8

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 D14 D14 D4×D7 D4×D7 D7×C4○D4 kernel D7×C4⋊D4 D7×C22⋊C4 D14⋊D4 D7×C4⋊C4 C4⋊D28 C28⋊7D4 C28⋊2D4 Dic7⋊D4 C7×C4⋊D4 D7×C22×C4 C2×D4×D7 C4×D7 C22×D7 C4⋊D4 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C22 C2 # reps 1 2 2 1 1 1 1 2 1 1 3 4 4 3 4 6 3 3 9 6 6 6

Matrix representation of D7×C4⋊D4 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 26 1 0 0 0 0 23 21 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 21 28 0 0 0 0 5 8 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 15 9 0 0 0 0 20 14 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 20 14 0 0 0 0 15 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 24 0 0 0 0 12 1
,
 9 15 0 0 0 0 14 20 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 24 0 0 0 0 0 1

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,23,0,0,0,0,1,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,21,5,0,0,0,0,28,8,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[15,20,0,0,0,0,9,14,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,15,0,0,0,0,14,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,12,0,0,0,0,24,1],[9,14,0,0,0,0,15,20,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,24,1] >;

D7×C4⋊D4 in GAP, Magma, Sage, TeX

D_7\times C_4\rtimes D_4
% in TeX

G:=Group("D7xC4:D4");
// GroupNames label

G:=SmallGroup(448,1057);
// by ID

G=gap.SmallGroup(448,1057);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,794,297,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^4=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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