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## G = D7×C2×C8order 224 = 25·7

### Direct product of C2×C8 and D7

Series: Derived Chief Lower central Upper central

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

Generators and relations for D7×C2×C8
G = < a,b,c,d | a2=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 238 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, C2×C14, C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, D7×C2×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, D14, C22×C8, C4×D7, C22×D7, C8×D7, C2×C4×D7, D7×C2×C8

Smallest permutation representation of D7×C2×C8
On 112 points
Generators in S112
(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)
(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 83 94 103 64 68 79)(2 84 95 104 57 69 80)(3 85 96 97 58 70 73)(4 86 89 98 59 71 74)(5 87 90 99 60 72 75)(6 88 91 100 61 65 76)(7 81 92 101 62 66 77)(8 82 93 102 63 67 78)(9 18 29 33 44 107 56)(10 19 30 34 45 108 49)(11 20 31 35 46 109 50)(12 21 32 36 47 110 51)(13 22 25 37 48 111 52)(14 23 26 38 41 112 53)(15 24 27 39 42 105 54)(16 17 28 40 43 106 55)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 73)(8 74)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 21)(18 22)(19 23)(20 24)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(41 108)(42 109)(43 110)(44 111)(45 112)(46 105)(47 106)(48 107)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 89)(64 90)(65 84)(66 85)(67 86)(68 87)(69 88)(70 81)(71 82)(72 83)(97 101)(98 102)(99 103)(100 104)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D7 D14 D14 C4×D7 C4×D7 C8×D7 kernel D7×C2×C8 C8×D7 C2×C7⋊C8 C2×C56 C2×C4×D7 C4×D7 C2×Dic7 C22×D7 D14 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 3 6 3 6 6 24

Matrix representation of D7×C2×C8 in GL3(𝔽113) generated by

 112 0 0 0 1 0 0 0 1
,
 112 0 0 0 69 0 0 0 69
,
 1 0 0 0 112 1 0 32 80
,
 112 0 0 0 1 0 0 81 112
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[112,0,0,0,69,0,0,0,69],[1,0,0,0,112,32,0,1,80],[112,0,0,0,1,81,0,0,112] >;

D7×C2×C8 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_8
% in TeX

G:=Group("D7xC2xC8");
// GroupNames label

G:=SmallGroup(224,94);
// by ID

G=gap.SmallGroup(224,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,50,69,6917]);
// Polycyclic

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

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