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## G = C7×D4○D8order 448 = 26·7

### Direct product of C7 and D4○D8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C7×D4○D8
G = < a,b,c,d,e | a7=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, D4○D8, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×Q16, D4×C14, D4×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, C7×C8○D4, C14×D8, C7×C4○D8, C7×C8⋊C22, C7×2+ 1+4, C7×D4○D8
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, D4○D8, D4×C14, C23×C14, D4×C2×C14, C7×D4○D8

Smallest permutation representation of C7×D4○D8
On 112 points
Generators in S112
(1 82 99 109 36 31 61)(2 83 100 110 37 32 62)(3 84 101 111 38 25 63)(4 85 102 112 39 26 64)(5 86 103 105 40 27 57)(6 87 104 106 33 28 58)(7 88 97 107 34 29 59)(8 81 98 108 35 30 60)(9 46 76 17 54 95 68)(10 47 77 18 55 96 69)(11 48 78 19 56 89 70)(12 41 79 20 49 90 71)(13 42 80 21 50 91 72)(14 43 73 22 51 92 65)(15 44 74 23 52 93 66)(16 45 75 24 53 94 67)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(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 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 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 97)(24 98)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 92)(34 93)(35 94)(36 95)(37 96)(38 89)(39 90)(40 91)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(73 87)(74 88)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)
(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 44)(2 43)(3 42)(4 41)(5 48)(6 47)(7 46)(8 45)(9 59)(10 58)(11 57)(12 64)(13 63)(14 62)(15 61)(16 60)(17 97)(18 104)(19 103)(20 102)(21 101)(22 100)(23 99)(24 98)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(49 112)(50 111)(51 110)(52 109)(53 108)(54 107)(55 106)(56 105)(73 83)(74 82)(75 81)(76 88)(77 87)(78 86)(79 85)(80 84)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E ··· 2J 4A 4B 4C 4D 4E 4F 7A ··· 7F 8A 8B 8C 8D 8E 14A ··· 14F 14G ··· 14X 14Y ··· 14BH 28A ··· 28X 28Y ··· 28AJ 56A ··· 56L 56M ··· 56AD order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 7 ··· 7 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 2 2 4 ··· 4 2 2 2 2 4 4 1 ··· 1 2 2 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 C7×D4 C7×D4 D4○D8 C7×D4○D8 kernel C7×D4○D8 C7×C8○D4 C14×D8 C7×C4○D8 C7×C8⋊C22 C7×2+ 1+4 D4○D8 C8○D4 C2×D8 C4○D8 C8⋊C22 2+ 1+4 C7×D4 C7×Q8 D4 Q8 C7 C1 # reps 1 1 3 3 6 2 6 6 18 18 36 12 3 1 18 6 2 12

Matrix representation of C7×D4○D8 in GL4(𝔽113) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 112 0 0 1 0 0 0 0 0 0 1 0 0 112 0
,
 0 0 0 1 0 0 112 0 0 112 0 0 1 0 0 0
,
 82 31 0 0 82 82 0 0 0 0 82 31 0 0 82 82
,
 0 0 1 0 0 0 0 112 1 0 0 0 0 112 0 0
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,112,0,0,0,0,0,0,112,0,0,1,0],[0,0,0,1,0,0,112,0,0,112,0,0,1,0,0,0],[82,82,0,0,31,82,0,0,0,0,82,82,0,0,31,82],[0,0,1,0,0,0,0,112,1,0,0,0,0,112,0,0] >;

C7×D4○D8 in GAP, Magma, Sage, TeX

C_7\times D_4\circ D_8
% in TeX

G:=Group("C7xD4oD8");
// GroupNames label

G:=SmallGroup(448,1359);
// by ID

G=gap.SmallGroup(448,1359);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1641,14117,7068,124]);
// Polycyclic

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

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