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## G = D7×C8○D4order 448 = 26·7

### Direct product of D7 and C8○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×C8○D4
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — D7×C4○D4 — D7×C8○D4
 Lower central C7 — C14 — D7×C8○D4
 Upper central C1 — C8 — C8○D4

Generators and relations for D7×C8○D4
G = < a,b,c,d,e | a7=b2=c8=e2=1, d2=c4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >

Subgroups: 956 in 266 conjugacy classes, 149 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, D14, C2×C14, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C7⋊C8, C7⋊C8, C56, C56, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C2×C8○D4, C8×D7, C8×D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, D7×C2×C8, D28.2C4, D7×M4(2), D28.C4, Q8.Dic7, C7×C8○D4, D7×C4○D4, D7×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, D14, C8○D4, C23×C4, C4×D7, C22×D7, C2×C8○D4, C2×C4×D7, C23×D7, D7×C22×C4, D7×C8○D4

Smallest permutation representation of D7×C8○D4
On 112 points
Generators in S112
(1 83 101 67 78 89 57)(2 84 102 68 79 90 58)(3 85 103 69 80 91 59)(4 86 104 70 73 92 60)(5 87 97 71 74 93 61)(6 88 98 72 75 94 62)(7 81 99 65 76 95 63)(8 82 100 66 77 96 64)(9 27 112 53 21 37 48)(10 28 105 54 22 38 41)(11 29 106 55 23 39 42)(12 30 107 56 24 40 43)(13 31 108 49 17 33 44)(14 32 109 50 18 34 45)(15 25 110 51 19 35 46)(16 26 111 52 20 36 47)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 110)(26 111)(27 112)(28 105)(29 106)(30 107)(31 108)(32 109)(73 104)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
(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)
(1 108 5 112)(2 109 6 105)(3 110 7 106)(4 111 8 107)(9 89 13 93)(10 90 14 94)(11 91 15 95)(12 92 16 96)(17 97 21 101)(18 98 22 102)(19 99 23 103)(20 100 24 104)(25 63 29 59)(26 64 30 60)(27 57 31 61)(28 58 32 62)(33 71 37 67)(34 72 38 68)(35 65 39 69)(36 66 40 70)(41 79 45 75)(42 80 46 76)(43 73 47 77)(44 74 48 78)(49 87 53 83)(50 88 54 84)(51 81 55 85)(52 82 56 86)
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 101)(18 102)(19 103)(20 104)(21 97)(22 98)(23 99)(24 100)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)

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

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

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

100 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 ··· 8J 8K 8L 8M 8N 8O ··· 8T 14A 14B 14C 14D ··· 14L 28A ··· 28F 28G ··· 28O 56A ··· 56L 56M ··· 56AD 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 8 8 8 ··· 8 14 14 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 2 2 7 7 14 14 14 1 1 2 2 2 7 7 14 14 14 2 2 2 1 1 1 1 2 ··· 2 7 7 7 7 14 ··· 14 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D7 D14 D14 D14 C8○D4 C4×D7 C4×D7 D7×C8○D4 kernel D7×C8○D4 D7×C2×C8 D28.2C4 D7×M4(2) D28.C4 Q8.Dic7 C7×C8○D4 D7×C4○D4 D4×D7 D4⋊2D7 Q8×D7 Q8⋊2D7 C8○D4 C2×C8 M4(2) C4○D4 D7 D4 Q8 C1 # reps 1 3 3 3 3 1 1 1 6 6 2 2 3 9 9 3 8 18 6 12

Matrix representation of D7×C8○D4 in GL4(𝔽113) generated by

 33 1 0 0 111 89 0 0 0 0 1 0 0 0 0 1
,
 104 9 0 0 79 9 0 0 0 0 1 0 0 0 0 1
,
 15 0 0 0 0 15 0 0 0 0 69 0 0 0 0 69
,
 112 0 0 0 0 112 0 0 0 0 0 112 0 0 1 0
,
 112 0 0 0 0 112 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(113))| [33,111,0,0,1,89,0,0,0,0,1,0,0,0,0,1],[104,79,0,0,9,9,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,15,0,0,0,0,69,0,0,0,0,69],[112,0,0,0,0,112,0,0,0,0,0,1,0,0,112,0],[112,0,0,0,0,112,0,0,0,0,0,1,0,0,1,0] >;

D7×C8○D4 in GAP, Magma, Sage, TeX

D_7\times C_8\circ D_4
% in TeX

G:=Group("D7xC8oD4");
// GroupNames label

G:=SmallGroup(448,1202);
// by ID

G=gap.SmallGroup(448,1202);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,80,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^8=e^2=1,d^2=c^4,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^4*d>;
// generators/relations

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