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## G = C2×D7×D8order 448 = 26·7

### Direct product of C2, D7 and D8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D7×D8
G = < a,b,c,d,e | a2=b7=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2212 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, C23, D7, D7, C14, C14, C14, C2×C8, C2×C8, D8, D8, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C22×C8, C2×D8, C2×D8, C22×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C22×D8, C8×D7, D56, C2×C7⋊C8, D4⋊D7, C2×C56, C7×D8, C2×C4×D7, C2×D28, D4×D7, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, D7×C2×C8, C2×D56, D7×D8, C2×D4⋊D7, C14×D8, C2×D4×D7, C2×D7×D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C24, D14, C2×D8, C22×D4, C22×D7, C22×D8, D4×D7, C23×D7, D7×D8, C2×D4×D7, C2×D7×D8

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D8 D14 D14 D14 D4×D7 D4×D7 D7×D8 kernel C2×D7×D8 D7×C2×C8 C2×D56 D7×D8 C2×D4⋊D7 C14×D8 C2×D4×D7 C4×D7 C2×Dic7 C22×D7 C2×D8 D14 C2×C8 D8 C2×D4 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 3 8 3 12 6 3 3 12

Matrix representation of C2×D7×D8 in GL5(𝔽113)

 112 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 112 0 0 0 0 0 112
,
 1 0 0 0 0 0 88 1 0 0 0 111 104 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 34 79 0 0 0 24 79 0 0 0 0 0 1 0 0 0 0 0 1
,
 112 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 51 0 0 0 31 62
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 62 0 0 0 31 0

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,88,111,0,0,0,1,104,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,24,0,0,0,79,79,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,51,62],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,62,0] >;

C2×D7×D8 in GAP, Magma, Sage, TeX

C_2\times D_7\times D_8
% in TeX

G:=Group("C2xD7xD8");
// GroupNames label

G:=SmallGroup(448,1207);
// by ID

G=gap.SmallGroup(448,1207);
# by ID

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

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

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