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## G = C7×D4.3D4order 448 = 26·7

### Direct product of C7 and D4.3D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×D4.3D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — D4×C14 — C14×SD16 — C7×D4.3D4
 Lower central C1 — C2 — C2×C4 — C7×D4.3D4
 Upper central C1 — C14 — C2×C28 — C7×D4.3D4

Generators and relations for C7×D4.3D4
G = < a,b,c,d,e | a7=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >

Subgroups: 194 in 104 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, D4.3D4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×D8, C7×SD16, C7×Q16, D4×C14, Q8×C14, C7×C4○D4, C7×C4.D4, C7×C4.10D4, C7×C8.C4, C7×C8○D4, C14×SD16, C7×C8⋊C22, C7×C8.C22, C7×D4.3D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C7×D4, C22×C14, D4.3D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×D4.3D4

Smallest permutation representation of C7×D4.3D4
On 112 points
Generators in S112
(1 25 43 73 91 17 35)(2 26 44 74 92 18 36)(3 27 45 75 93 19 37)(4 28 46 76 94 20 38)(5 29 47 77 95 21 39)(6 30 48 78 96 22 40)(7 31 41 79 89 23 33)(8 32 42 80 90 24 34)(9 108 68 52 82 100 63)(10 109 69 53 83 101 64)(11 110 70 54 84 102 57)(12 111 71 55 85 103 58)(13 112 72 56 86 104 59)(14 105 65 49 87 97 60)(15 106 66 50 88 98 61)(16 107 67 51 81 99 62)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 79 77 75)(74 80 78 76)(81 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
(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 4 5 8)(2 7 6 3)(9 16 13 12)(10 11 14 15)(17 20 21 24)(18 23 22 19)(25 28 29 32)(26 31 30 27)(33 40 37 36)(34 35 38 39)(41 48 45 44)(42 43 46 47)(49 50 53 54)(51 56 55 52)(57 60 61 64)(58 63 62 59)(65 66 69 70)(67 72 71 68)(73 76 77 80)(74 79 78 75)(81 86 85 82)(83 84 87 88)(89 96 93 92)(90 91 94 95)(97 98 101 102)(99 104 103 100)(105 106 109 110)(107 112 111 108)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 8E 8F 8G 14A ··· 14F 14G ··· 14L 14M ··· 14R 14S ··· 14X 28A ··· 28L 28M ··· 28R 28S ··· 28X 56A ··· 56L 56M ··· 56AD 56AE ··· 56AP order 1 2 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 56 ··· 56 size 1 1 2 4 8 2 2 4 8 1 ··· 1 2 2 4 4 4 8 8 1 ··· 1 2 ··· 2 4 ··· 4 8 ··· 8 2 ··· 2 4 ··· 4 8 ··· 8 2 ··· 2 4 ··· 4 8 ··· 8

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C14 D4 D4 D4 C4○D4 C7×D4 C7×D4 C7×D4 C7×C4○D4 D4.3D4 C7×D4.3D4 kernel C7×D4.3D4 C7×C4.D4 C7×C4.10D4 C7×C8.C4 C7×C8○D4 C14×SD16 C7×C8⋊C22 C7×C8.C22 D4.3D4 C4.D4 C4.10D4 C8.C4 C8○D4 C2×SD16 C8⋊C22 C8.C22 C56 C7×D4 C7×Q8 C2×C14 C8 D4 Q8 C22 C7 C1 # reps 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 2 1 1 2 12 6 6 12 2 12

Matrix representation of C7×D4.3D4 in GL4(𝔽113) generated by

 106 0 0 0 0 106 0 0 0 0 106 0 0 0 0 106
,
 1 111 0 0 1 112 0 0 112 1 0 112 0 1 1 0
,
 1 0 0 2 1 0 1 1 112 1 0 112 0 0 0 112
,
 87 26 0 0 100 0 0 0 0 100 100 100 13 100 13 100
,
 87 26 0 0 100 26 0 0 13 100 13 100 0 100 100 100
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[1,1,112,0,111,112,1,1,0,0,0,1,0,0,112,0],[1,1,112,0,0,0,1,0,0,1,0,0,2,1,112,112],[87,100,0,13,26,0,100,100,0,0,100,13,0,0,100,100],[87,100,13,0,26,26,100,100,0,0,13,100,0,0,100,100] >;

C7×D4.3D4 in GAP, Magma, Sage, TeX

C_7\times D_4._3D_4
% in TeX

G:=Group("C7xD4.3D4");
// GroupNames label

G:=SmallGroup(448,879);
// by ID

G=gap.SmallGroup(448,879);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,400,2438,9804,172,14117,3547,124]);
// Polycyclic

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

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