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## G = D7×C22⋊C8order 448 = 26·7

### Direct product of D7 and C22⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×C22⋊C8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C22×C4 — D7×C22⋊C8
 Lower central C7 — C14 — D7×C22⋊C8
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for D7×C22⋊C8
G = < a,b,c,d,e | a7=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 956 in 202 conjugacy classes, 73 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C23, D7, D7, C14, C14, C2×C8, C2×C8, C22×C4, C22×C4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, C22⋊C8, C22×C8, C23×C4, C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×D7, C22×C14, C2×C22⋊C8, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D14⋊C8, C28.55D4, C7×C22⋊C8, D7×C2×C8, D7×C22×C4, D7×C22⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D7, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, D14, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C4×D7, C22×D7, C2×C22⋊C8, C8×D7, C2×C4×D7, D4×D7, D7×C22⋊C4, D7×C2×C8, D7×M4(2), D7×C22⋊C8

Smallest permutation representation of D7×C22⋊C8
On 112 points
Generators in S112
(1 71 14 63 88 20 97)(2 72 15 64 81 21 98)(3 65 16 57 82 22 99)(4 66 9 58 83 23 100)(5 67 10 59 84 24 101)(6 68 11 60 85 17 102)(7 69 12 61 86 18 103)(8 70 13 62 87 19 104)(25 112 44 90 38 79 54)(26 105 45 91 39 80 55)(27 106 46 92 40 73 56)(28 107 47 93 33 74 49)(29 108 48 94 34 75 50)(30 109 41 95 35 76 51)(31 110 42 96 36 77 52)(32 111 43 89 37 78 53)
(1 109)(2 110)(3 111)(4 112)(5 105)(6 106)(7 107)(8 108)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 92)(18 93)(19 94)(20 95)(21 96)(22 89)(23 90)(24 91)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 65)(33 86)(34 87)(35 88)(36 81)(37 82)(38 83)(39 84)(40 85)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(57 78)(58 79)(59 80)(60 73)(61 74)(62 75)(63 76)(64 77)
(1 5)(2 42)(3 7)(4 44)(6 46)(8 48)(9 38)(10 14)(11 40)(12 16)(13 34)(15 36)(17 27)(18 22)(19 29)(20 24)(21 31)(23 25)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(50 87)(51 55)(52 81)(54 83)(56 85)(57 61)(58 79)(59 63)(60 73)(62 75)(64 77)(65 69)(66 90)(67 71)(68 92)(70 94)(72 96)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(97 101)(98 110)(99 103)(100 112)(102 106)(104 108)(105 109)(107 111)
(1 45)(2 46)(3 47)(4 48)(5 41)(6 42)(7 43)(8 44)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 81)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 73)(65 93)(66 94)(67 95)(68 96)(69 89)(70 90)(71 91)(72 92)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)
(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)

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

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

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

100 conjugacy classes

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

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 D4 D7 M4(2) D14 D14 C4×D7 C4×D7 C8×D7 D4×D7 D7×M4(2) kernel D7×C22⋊C8 D14⋊C8 C28.55D4 C7×C22⋊C8 D7×C2×C8 D7×C22×C4 C2×C4×D7 C22×Dic7 C23×D7 C22×D7 C4×D7 C22⋊C8 D14 C2×C8 C22×C4 C2×C4 C23 C22 C4 C2 # reps 1 2 1 1 2 1 4 2 2 16 4 3 4 6 3 6 6 24 6 6

Matrix representation of D7×C22⋊C8 in GL4(𝔽113) generated by

 0 1 0 0 112 79 0 0 0 0 1 0 0 0 0 1
,
 0 112 0 0 112 0 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 112 112
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 95 0 0 0 0 95 0 0 0 0 112 111 0 0 0 1
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,112,0,0,112,0,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,112,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[95,0,0,0,0,95,0,0,0,0,112,0,0,0,111,1] >;

D7×C22⋊C8 in GAP, Magma, Sage, TeX

D_7\times C_2^2\rtimes C_8
% in TeX

G:=Group("D7xC2^2:C8");
// GroupNames label

G:=SmallGroup(448,258);
// by ID

G=gap.SmallGroup(448,258);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,58,136,18822]);
// Polycyclic

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

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