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## G = D7×C4○D8order 448 = 26·7

### Direct product of D7 and C4○D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×C4○D8
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — D7×C4○D4 — D7×C4○D8
 Lower central C7 — C14 — C28 — D7×C4○D8
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for D7×C4○D8
G = < a,b,c,d,e | a7=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >

Subgroups: 1332 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, C2×C8, D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C2×C4○D8, C8×D7, C56⋊C2, D56, Dic28, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, D7×C2×C8, D567C2, D7×D8, D83D7, D7×SD16, SD163D7, D7×Q16, Q8.D14, D4.8D14, C7×C4○D8, D7×C4○D4, D7×C4○D8
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4○D8, C22×D4, C22×D7, C2×C4○D8, D4×D7, C23×D7, C2×D4×D7, D7×C4○D8

Smallest permutation representation of D7×C4○D8
On 112 points
Generators in S112
(1 102 14 111 93 40 57)(2 103 15 112 94 33 58)(3 104 16 105 95 34 59)(4 97 9 106 96 35 60)(5 98 10 107 89 36 61)(6 99 11 108 90 37 62)(7 100 12 109 91 38 63)(8 101 13 110 92 39 64)(17 84 29 74 50 66 41)(18 85 30 75 51 67 42)(19 86 31 76 52 68 43)(20 87 32 77 53 69 44)(21 88 25 78 54 70 45)(22 81 26 79 55 71 46)(23 82 27 80 56 72 47)(24 83 28 73 49 65 48)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 96)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 88)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 103)(34 104)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)
(1 3 5 7)(2 4 6 8)(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 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 71 69 67)(66 72 70 68)(73 79 77 75)(74 80 78 76)(81 87 85 83)(82 88 86 84)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 50)(10 49)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 35)(18 34)(19 33)(20 40)(21 39)(22 38)(23 37)(24 36)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 88)(65 107)(66 106)(67 105)(68 112)(69 111)(70 110)(71 109)(72 108)(73 98)(74 97)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 C4○D8 D4×D7 D4×D7 D7×C4○D8 kernel D7×C4○D8 D7×C2×C8 D56⋊7C2 D7×D8 D8⋊3D7 D7×SD16 SD16⋊3D7 D7×Q16 Q8.D14 D4.8D14 C7×C4○D8 D7×C4○D4 C4×D7 C2×Dic7 C22×D7 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 D7 C4 C22 C1 # reps 1 1 1 1 1 2 2 1 1 2 1 2 2 1 1 3 3 3 6 3 6 8 3 3 12

Matrix representation of D7×C4○D8 in GL4(𝔽113) generated by

 0 1 0 0 112 9 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 98 0 0 0 0 98
,
 1 0 0 0 0 1 0 0 0 0 95 0 0 0 100 69
,
 1 0 0 0 0 1 0 0 0 0 44 88 0 0 100 69
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,95,100,0,0,0,69],[1,0,0,0,0,1,0,0,0,0,44,100,0,0,88,69] >;

D7×C4○D8 in GAP, Magma, Sage, TeX

D_7\times C_4\circ D_8
% in TeX

G:=Group("D7xC4oD8");
// GroupNames label

G:=SmallGroup(448,1220);
// by ID

G=gap.SmallGroup(448,1220);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,570,185,438,235,102,18822]);
// Polycyclic

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

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