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## G = (D4×D7)⋊C4order 448 = 26·7

### 2nd semidirect product of D4×D7 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — (D4×D7)⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C2×D4×D7 — (D4×D7)⋊C4
 Lower central C7 — C14 — C28 — (D4×D7)⋊C4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for (D4×D7)⋊C4
G = < a,b,c,d,e | a4=b2=c7=d2=e4=1, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >

Subgroups: 1268 in 190 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, D4⋊C4, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C23.37D4, C8⋊D7, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, D4×D7, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C14.D8, C2.D56, D4⋊Dic7, C7×D4⋊C4, C4⋊C47D7, C2×C8⋊D7, C2×D4×D7, (D4×D7)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8⋊C22, C4×D7, C22×D7, C23.37D4, C2×C4×D7, D4×D7, D7×C22⋊C4, D8⋊D7, D56⋊C2, (D4×D7)⋊C4

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 14 14 28 28 2 2 4 4 14 14 28 28 2 2 2 4 4 28 28 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D7 D14 D14 D14 C4×D7 C8⋊C22 D4×D7 D4×D7 D8⋊D7 D56⋊C2 kernel (D4×D7)⋊C4 C14.D8 C2.D56 D4⋊Dic7 C7×D4⋊C4 C4⋊C4⋊7D7 C2×C8⋊D7 C2×D4×D7 D4×D7 C4×D7 C2×Dic7 C22×D7 D4⋊C4 C4⋊C4 C2×C8 C2×D4 D4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 3 3 3 3 12 2 3 3 6 6

Matrix representation of (D4×D7)⋊C4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 70 0 0 0 112 43 58 0 0 33 42 1 0 0 0 71 0 0 1
,
 1 0 0 0 0 0 8 112 0 0 0 0 0 0 1 0 0 43 0 0 0 1 70 55 0 0 0 0 112 0 0 0 0 0 0 112
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 88 112 0 0 0 0 2 104 0 0 0 0 0 0 79 112 0 0 0 0 1 0
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 26 25 0 0 0 0 86 87 0 0 0 0 0 0 34 1 0 0 0 0 88 79
,
 7 83 0 0 0 0 77 106 0 0 0 0 0 0 103 27 41 19 0 0 59 10 11 18 0 0 56 79 106 86 0 0 16 2 27 7

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,33,71,0,0,0,112,42,0,0,0,0,43,1,0,0,0,70,58,0,1],[1,8,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,70,112,0,0,0,43,55,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,2,0,0,0,0,112,104,0,0,0,0,0,0,79,1,0,0,0,0,112,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,26,86,0,0,0,0,25,87,0,0,0,0,0,0,34,88,0,0,0,0,1,79],[7,77,0,0,0,0,83,106,0,0,0,0,0,0,103,59,56,16,0,0,27,10,79,2,0,0,41,11,106,27,0,0,19,18,86,7] >;

(D4×D7)⋊C4 in GAP, Magma, Sage, TeX

(D_4\times D_7)\rtimes C_4
% in TeX

G:=Group("(D4xD7):C4");
// GroupNames label

G:=SmallGroup(448,304);
// by ID

G=gap.SmallGroup(448,304);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,58,851,438,102,18822]);
// Polycyclic

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

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