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## G = C2×D8⋊D7order 448 = 26·7

### Direct product of C2 and D8⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D8⋊D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×D4×D7 — C2×D8⋊D7
 Lower central C7 — C14 — C28 — C2×D8⋊D7
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for C2×D8⋊D7
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1700 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C2×C8⋊C22, C8⋊D7, C56⋊C2, C2×C7⋊C8, D4⋊D7, D4.D7, C2×C56, C7×D8, C2×Dic14, C2×C4×D7, C2×D28, D4×D7, D4×D7, D42D7, D42D7, C22×Dic7, C2×C7⋊D4, D4×C14, C23×D7, C2×C8⋊D7, C2×C56⋊C2, D8⋊D7, C2×D4⋊D7, C2×D4.D7, C14×D8, C2×D4×D7, C2×D42D7, C2×D8⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8⋊C22, C22×D4, C22×D7, C2×C8⋊C22, D4×D7, C23×D7, D8⋊D7, C2×D4×D7, C2×D8⋊D7

Smallest permutation representation of C2×D8⋊D7
On 112 points
Generators in S112
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 100)(10 101)(11 102)(12 103)(13 104)(14 97)(15 98)(16 99)(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 65)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(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)
(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)
(1 106)(2 105)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 68)(26 67)(27 66)(28 65)(29 72)(30 71)(31 70)(32 69)(33 60)(34 59)(35 58)(36 57)(37 64)(38 63)(39 62)(40 61)(41 75)(42 74)(43 73)(44 80)(45 79)(46 78)(47 77)(48 76)(49 81)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)
(1 72 61 99 96 83 80)(2 65 62 100 89 84 73)(3 66 63 101 90 85 74)(4 67 64 102 91 86 75)(5 68 57 103 92 87 76)(6 69 58 104 93 88 77)(7 70 59 97 94 81 78)(8 71 60 98 95 82 79)(9 20 50 47 109 32 35)(10 21 51 48 110 25 36)(11 22 52 41 111 26 37)(12 23 53 42 112 27 38)(13 24 54 43 105 28 39)(14 17 55 44 106 29 40)(15 18 56 45 107 30 33)(16 19 49 46 108 31 34)
(1 46)(2 43)(3 48)(4 45)(5 42)(6 47)(7 44)(8 41)(9 104)(10 101)(11 98)(12 103)(13 100)(14 97)(15 102)(16 99)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(25 85)(26 82)(27 87)(28 84)(29 81)(30 86)(31 83)(32 88)(33 91)(34 96)(35 93)(36 90)(37 95)(38 92)(39 89)(40 94)(49 72)(50 69)(51 66)(52 71)(53 68)(54 65)(55 70)(56 67)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 C8⋊C22 D4×D7 D4×D7 D8⋊D7 kernel C2×D8⋊D7 C2×C8⋊D7 C2×C56⋊C2 D8⋊D7 C2×D4⋊D7 C2×D4.D7 C14×D8 C2×D4×D7 C2×D4⋊2D7 C4×D7 C2×Dic7 C22×D7 C2×D8 C2×C8 D8 C2×D4 C14 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 3 3 12 6 2 3 3 12

Matrix representation of C2×D8⋊D7 in GL8(𝔽113)

 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 0 0 0 91 28 0 0 0 0 0 0 58 69 0 0 0 0 2 104 105 84 0 0 0 0 54 1 45 8
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 80 71 0 0 0 0 0 112 42 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 79 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 79 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9 112 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 88 112 0 0 0 0 0 0 60 34
,
 34 1 0 0 0 0 0 0 88 79 0 0 0 0 0 0 0 0 34 1 0 0 0 0 0 0 88 79 0 0 0 0 0 0 0 0 9 112 0 0 0 0 0 0 80 104 0 0 0 0 0 0 0 0 25 33 0 0 0 0 0 0 53 88

G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,54,0,0,0,0,0,0,104,1,0,0,0,0,91,58,105,45,0,0,0,0,28,69,84,8],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,80,42,1,0,0,0,0,0,71,0,0,1],[79,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,79,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,88,60,0,0,0,0,0,0,112,34],[34,88,0,0,0,0,0,0,1,79,0,0,0,0,0,0,0,0,34,88,0,0,0,0,0,0,1,79,0,0,0,0,0,0,0,0,9,80,0,0,0,0,0,0,112,104,0,0,0,0,0,0,0,0,25,53,0,0,0,0,0,0,33,88] >;

C2×D8⋊D7 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes D_7
% in TeX

G:=Group("C2xD8:D7");
// GroupNames label

G:=SmallGroup(448,1208);
// by ID

G=gap.SmallGroup(448,1208);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1123,185,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,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,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
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