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G = D8:2Dic7order 448 = 26·7

2nd semidirect product of D8 and Dic7 acting via Dic7/C14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:2Dic7, C56.40D4, Q16:2Dic7, C28.37SD16, (C7xD8):4C4, (C7xQ16):4C4, C4oD8.2D7, (C2xC14).4D8, C7:3(D8:2C4), C56.27(C2xC4), C8:Dic7:23C2, (C2xC8).51D14, C28.C8:8C2, C8.3(C2xDic7), (C2xC28).118D4, C8.30(C7:D4), C4.12(D4.D7), C22.3(D4:D7), C4.5(C23.D7), C28.17(C22:C4), (C2xC56).154C22, C14.30(D4:C4), C2.10(D4:Dic7), (C7xC4oD8).5C2, (C2xC4).26(C7:D4), SmallGroup(448,123)

Series: Derived Chief Lower central Upper central

C1C56 — D8:2Dic7
C1C7C14C28C2xC28C2xC56C8:Dic7 — D8:2Dic7
C7C14C28C56 — D8:2Dic7
C1C2C2xC4C2xC8C4oD8

Generators and relations for D8:2Dic7
 G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 244 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, Q8, C14, C14, C16, C4:C4, C2xC8, D8, SD16, Q16, C4oD4, Dic7, C28, C28, C2xC14, C2xC14, C4.Q8, M5(2), C4oD8, C56, C2xDic7, C2xC28, C2xC28, C7xD4, C7xQ8, D8:2C4, C7:C16, C4:Dic7, C2xC56, C7xD8, C7xSD16, C7xQ16, C7xC4oD4, C28.C8, C8:Dic7, C7xC4oD8, D8:2Dic7
Quotients: C1, C2, C4, C22, C2xC4, D4, D7, C22:C4, D8, SD16, Dic7, D14, D4:C4, C2xDic7, C7:D4, D8:2C4, D4:D7, D4.D7, C23.D7, D4:Dic7, D8:2Dic7

Smallest permutation representation of D8:2Dic7
On 112 points
Generators in S112
(1 50 27 40 13 43 21 34)(2 51 28 41 14 44 15 35)(3 52 22 42 8 45 16 29)(4 53 23 36 9 46 17 30)(5 54 24 37 10 47 18 31)(6 55 25 38 11 48 19 32)(7 56 26 39 12 49 20 33)(57 98 84 103 64 91 77 110)(58 85 71 104 65 92 78 111)(59 86 72 105 66 93 79 112)(60 87 73 106 67 94 80 99)(61 88 74 107 68 95 81 100)(62 89 75 108 69 96 82 101)(63 90 76 109 70 97 83 102)
(1 100)(2 108)(3 102)(4 110)(5 104)(6 112)(7 106)(8 109)(9 103)(10 111)(11 105)(12 99)(13 107)(14 101)(15 96)(16 90)(17 98)(18 92)(19 86)(20 94)(21 88)(22 97)(23 91)(24 85)(25 93)(26 87)(27 95)(28 89)(29 63)(30 57)(31 65)(32 59)(33 67)(34 61)(35 69)(36 64)(37 58)(38 66)(39 60)(40 68)(41 62)(42 70)(43 74)(44 82)(45 76)(46 84)(47 78)(48 72)(49 80)(50 81)(51 75)(52 83)(53 77)(54 71)(55 79)(56 73)
(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)
(2 7)(3 6)(4 5)(8 11)(9 10)(12 14)(15 26)(16 25)(17 24)(18 23)(19 22)(20 28)(21 27)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 49)(36 54)(37 53)(38 52)(39 51)(40 50)(41 56)(42 55)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 103 78 110)(72 102 79 109)(73 101 80 108)(74 100 81 107)(75 99 82 106)(76 112 83 105)(77 111 84 104)

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

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

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

58 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A7B7C8A8B8C14A14B14C14D14E14F14G···14L16A16B16C16D28A···28F28G28H28I28J···28O56A···56L
order12224444477788814141414141414···141616161628···2828282828···2856···56
size112822856562222242224448···8282828282···24448···84···4

58 irreducible representations

dim11111122222222224444
type+++++++++---+
imageC1C2C2C2C4C4D4D4D7SD16D8D14Dic7Dic7C7:D4C7:D4D8:2C4D4.D7D4:D7D8:2Dic7
kernelD8:2Dic7C28.C8C8:Dic7C7xC4oD8C7xD8C7xQ16C56C2xC28C4oD8C28C2xC14C2xC8D8Q16C8C2xC4C7C4C22C1
# reps111122113223336623312

Matrix representation of D8:2Dic7 in GL4(F113) generated by

1047100
349600
23198971
9404250
,
96421113
791046536
23198971
9404250
,
34100
538800
7100112
41711104
,
010400
25000
171015042
98274063
G:=sub<GL(4,GF(113))| [104,34,23,94,71,96,19,0,0,0,89,42,0,0,71,50],[96,79,23,94,42,104,19,0,111,65,89,42,3,36,71,50],[34,53,71,41,1,88,0,71,0,0,0,1,0,0,112,104],[0,25,17,98,104,0,101,27,0,0,50,40,0,0,42,63] >;

D8:2Dic7 in GAP, Magma, Sage, TeX

D_8\rtimes_2{\rm Dic}_7
% in TeX

G:=Group("D8:2Dic7");
// GroupNames label

G:=SmallGroup(448,123);
// by ID

G=gap.SmallGroup(448,123);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,675,794,80,1684,851,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

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