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## G = D7×2- 1+4order 448 = 26·7

### Direct product of D7 and 2- 1+4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×2- 1+4
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — C2×Q8×D7 — D7×2- 1+4
 Lower central C7 — C14 — D7×2- 1+4
 Upper central C1 — C2 — 2- 1+4

Generators and relations for D7×2- 1+4
G = < a,b,c,d,e,f | a7=b2=c4=d2=1, e2=f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 2868 in 794 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, C28, D14, D14, D14, C2×C14, C22×Q8, C2×C4○D4, 2- 1+4, 2- 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C2×2- 1+4, C2×Dic14, C2×C4×D7, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, Q8×C14, C7×C4○D4, C2×Q8×D7, Q8.10D14, D7×C4○D4, D4.10D14, C7×2- 1+4, D7×2- 1+4
Quotients: C1, C2, C22, C23, D7, C24, D14, 2- 1+4, C25, C22×D7, C2×2- 1+4, C23×D7, D7×C24, D7×2- 1+4

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

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

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

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

85 conjugacy classes

 class 1 2A 2B ··· 2F 2G 2H 2I ··· 2M 4A ··· 4J 4K ··· 4T 7A 7B 7C 14A 14B 14C 14D ··· 14R 28A ··· 28AD order 1 2 2 ··· 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 14 14 14 ··· 14 28 ··· 28 size 1 1 2 ··· 2 7 7 14 ··· 14 2 ··· 2 14 ··· 14 2 2 2 2 2 2 4 ··· 4 4 ··· 4

85 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D7 D14 D14 2- 1+4 D7×2- 1+4 kernel D7×2- 1+4 C2×Q8×D7 Q8.10D14 D7×C4○D4 D4.10D14 C7×2- 1+4 2- 1+4 C2×Q8 C4○D4 D7 C1 # reps 1 5 5 10 10 1 3 15 30 2 3

Matrix representation of D7×2- 1+4 in GL6(𝔽29)

 0 1 0 0 0 0 28 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 28 0 0 0 0 28 0 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 14 0 16 26 0 0 0 27 0 16 0 0 4 3 15 3 0 0 0 16 0 2
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 28 0 0 0 1 0 1 28 0 0 0 0 1 0 0 0 28 28 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 14 0 16 26 0 0 15 2 13 16 0 0 26 26 11 3 0 0 11 13 27 2
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 15 4 13 0 0 0 0 27 0 16 0 0 25 25 14 4 0 0 0 16 0 2

G:=sub<GL(6,GF(29))| [0,28,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,14,0,4,0,0,0,0,27,3,16,0,0,16,0,15,0,0,0,26,16,3,2],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,1,0,28,0,0,0,0,0,28,0,0,28,1,1,0,0,0,0,28,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,15,26,11,0,0,0,2,26,13,0,0,16,13,11,27,0,0,26,16,3,2],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,15,0,25,0,0,0,4,27,25,16,0,0,13,0,14,0,0,0,0,16,4,2] >;

D7×2- 1+4 in GAP, Magma, Sage, TeX

D_7\times 2_-^{1+4}
% in TeX

G:=Group("D7xES-(2,2)");
// GroupNames label

G:=SmallGroup(448,1381);
// by ID

G=gap.SmallGroup(448,1381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,297,136,851,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^7=b^2=c^4=d^2=1,e^2=f^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c^2*e>;
// generators/relations

׿
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