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## G = D7×C8.C22order 448 = 26·7

### Direct product of D7 and C8.C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×C8.C22
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×Q8×D7 — D7×C8.C22
 Lower central C7 — C14 — C28 — D7×C8.C22
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D7×C8.C22
G = < a,b,c,d,e | a7=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >

Subgroups: 1228 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×D7, C22×D7, C2×C8.C22, C8×D7, C8⋊D7, C56⋊C2, Dic28, C4.Dic7, D4.D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q8×D7, Q8×D7, Q82D7, Q8×C14, C7×C4○D4, D7×M4(2), C8.D14, D7×SD16, SD16⋊D7, D7×Q16, Q16⋊D7, C28.C23, D4.9D14, C7×C8.C22, C2×Q8×D7, D7×C4○D4, D7×C8.C22
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8.C22, C22×D4, C22×D7, C2×C8.C22, D4×D7, C23×D7, C2×D4×D7, D7×C8.C22

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

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

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

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

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 C8.C22 D4×D7 D4×D7 D7×C8.C22 kernel D7×C8.C22 D7×M4(2) C8.D14 D7×SD16 SD16⋊D7 D7×Q16 Q16⋊D7 C28.C23 D4.9D14 C7×C8.C22 C2×Q8×D7 D7×C4○D4 C4×D7 C2×Dic7 C22×D7 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 D7 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 1 1 3 3 6 6 3 3 2 3 3 3

Matrix representation of D7×C8.C22 in GL8(𝔽113)

 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 112 0 9 0 0 0 0 0 0 112 0 9 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 0 112 0 0 0 0 0 0 0 0 112 0 0 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
,
 41 10 0 0 0 0 0 0 103 72 0 0 0 0 0 0 0 0 41 10 0 0 0 0 0 0 103 72 0 0 0 0 0 0 0 0 60 60 104 77 0 0 0 0 44 44 97 64 0 0 0 0 60 44 0 0 0 0 0 0 83 109 109 9
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 112 112 112 91 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 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 0 112 0 0 0 0 0 41 41 0 112

G:=sub<GL(8,GF(113))| [0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,9,0,0,0,0,0,0,1,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[41,103,0,0,0,0,0,0,10,72,0,0,0,0,0,0,0,0,41,103,0,0,0,0,0,0,10,72,0,0,0,0,0,0,0,0,60,44,60,83,0,0,0,0,60,44,44,109,0,0,0,0,104,97,0,109,0,0,0,0,77,64,0,9],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,0,0,0,0,0,0,91,0,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,41,0,0,0,0,0,1,0,41,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112] >;

D7×C8.C22 in GAP, Magma, Sage, TeX

D_7\times C_8.C_2^2
% in TeX

G:=Group("D7xC8.C2^2");
// GroupNames label

G:=SmallGroup(448,1229);
// by ID

G=gap.SmallGroup(448,1229);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,185,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^8=d^2=e^2=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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations

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