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G = D810D14order 448 = 26·7

4th semidirect product of D8 and D14 acting via D14/C14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D810D14, Q169D14, SD1610D14, D5622C22, C28.15C24, C56.37C23, D28.10C23, Dic2819C22, Dic14.10C23, C4○D83D7, C4○D48D14, (C2×C8)⋊12D14, C7⋊C8.6C23, D8⋊D77C2, D56⋊C27C2, (C2×C56)⋊5C22, (C4×D7).51D4, C4.222(D4×D7), (D4×D7)⋊7C22, D567C27C2, (Q8×D7)⋊8C22, C22.5(D4×D7), D4⋊D713C22, Q16⋊D77C2, D14.53(C2×D4), C28.381(C2×D4), SD16⋊D77C2, C4○D286C22, (C7×D8)⋊15C22, Q8⋊D712C22, D4.9(C22×D7), (C7×D4).9C23, (C4×D7).8C23, C8.15(C22×D7), C4.15(C23×D7), D4.8D142C2, Q8.9(C22×D7), (C7×Q8).9C23, C8⋊D715C22, C56⋊C216C22, C72(D8⋊C22), D4.D712C22, (C2×Dic7).80D4, Dic7.58(C2×D4), (C7×Q16)⋊13C22, (C22×D7).42D4, C7⋊Q1611C22, (C2×C28).532C23, (C7×SD16)⋊10C22, D42D7.5C22, C14.116(C22×D4), Q82D7.5C22, C2.89(C2×D4×D7), (C7×C4○D8)⋊3C2, (D7×C4○D4)⋊2C2, (C2×C8⋊D7)⋊1C2, (C2×C7⋊C8)⋊16C22, (C2×C14).12(C2×D4), (C7×C4○D4)⋊2C22, (C2×C4×D7).159C22, (C2×C4).619(C22×D7), SmallGroup(448,1221)

Series: Derived Chief Lower central Upper central

C1C28 — D810D14
C1C7C14C28C4×D7C2×C4×D7D7×C4○D4 — D810D14
C7C14C28 — D810D14
C1C4C2×C4C4○D8

Generators and relations for D810D14
 G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 1332 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C4○D8, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, D8⋊C22, C8⋊D7, C56⋊C2, D56, Dic28, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, C2×C8⋊D7, D567C2, D8⋊D7, D56⋊C2, SD16⋊D7, Q16⋊D7, D4.8D14, C7×C4○D8, D7×C4○D4, D810D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D8⋊C22, D4×D7, C23×D7, C2×D4×D7, D810D14

Smallest permutation representation of D810D14
On 112 points
Generators in S112
(1 51 38 16 46 9 33 23)(2 52 39 17 47 10 34 24)(3 53 40 18 48 11 35 25)(4 54 41 19 49 12 29 26)(5 55 42 20 43 13 30 27)(6 56 36 21 44 14 31 28)(7 50 37 15 45 8 32 22)(57 107 83 95 64 100 76 88)(58 108 84 96 65 101 77 89)(59 109 71 97 66 102 78 90)(60 110 72 98 67 103 79 91)(61 111 73 85 68 104 80 92)(62 112 74 86 69 105 81 93)(63 99 75 87 70 106 82 94)
(1 109)(2 103)(3 111)(4 105)(5 99)(6 107)(7 101)(8 58)(9 66)(10 60)(11 68)(12 62)(13 70)(14 64)(15 84)(16 78)(17 72)(18 80)(19 74)(20 82)(21 76)(22 77)(23 71)(24 79)(25 73)(26 81)(27 75)(28 83)(29 93)(30 87)(31 95)(32 89)(33 97)(34 91)(35 85)(36 88)(37 96)(38 90)(39 98)(40 92)(41 86)(42 94)(43 106)(44 100)(45 108)(46 102)(47 110)(48 104)(49 112)(50 65)(51 59)(52 67)(53 61)(54 69)(55 63)(56 57)
(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 44)(2 43)(3 49)(4 48)(5 47)(6 46)(7 45)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 40)(30 39)(31 38)(32 37)(33 36)(34 42)(35 41)(51 56)(52 55)(53 54)(57 66)(58 65)(59 64)(60 63)(61 62)(67 70)(68 69)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 93)(86 92)(87 91)(88 90)(94 98)(95 97)(99 103)(100 102)(104 112)(105 111)(106 110)(107 109)

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

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

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

64 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I7A7B7C8A8B8C8D14A14B14C14D14E14F14G···14L28A···28F28G28H28I28J···28O56A···56L
order122222222444444444777888814141414141414···1428···2828282828···2856···56
size112441414282811244141428282224428282224448···82···24448···84···4

64 irreducible representations

dim11111111112222222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4D7D14D14D14D14D14D8⋊C22D4×D7D4×D7D810D14
kernelD810D14C2×C8⋊D7D567C2D8⋊D7D56⋊C2SD16⋊D7Q16⋊D7D4.8D14C7×C4○D8D7×C4○D4C4×D7C2×Dic7C22×D7C4○D8C2×C8D8SD16Q16C4○D4C7C4C22C1
# reps111222221221133363623312

Matrix representation of D810D14 in GL6(𝔽113)

100000
010000
000001
000010
0001500
0098000
,
11200000
01120000
000010
000001
001000
000100
,
0800000
24890000
001000
00011200
00001120
000001
,
89330000
99240000
001000
000100
00001120
00000112

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,98,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1,0,0,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,24,0,0,0,0,80,89,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1],[89,99,0,0,0,0,33,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;

D810D14 in GAP, Magma, Sage, TeX

D_8\rtimes_{10}D_{14}
% in TeX

G:=Group("D8:10D14");
// GroupNames label

G:=SmallGroup(448,1221);
// by ID

G=gap.SmallGroup(448,1221);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,570,185,438,235,102,18822]);
// Polycyclic

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

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