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G = Q8⋊D26order 416 = 25·13

2nd semidirect product of Q8 and D26 acting via D26/D13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D26, Q82D26, D1046C2, D4.3D26, C1043C22, SD161D13, D26.15D4, D522C22, C52.5C23, Dic13.17D4, D4⋊D133C2, (D4×D13)⋊3C2, Q8⋊D132C2, C8⋊D131C2, C133(C8⋊C22), C2.19(D4×D13), C26.31(C2×D4), D52⋊C21C2, C132C82C22, (C13×SD16)⋊1C2, (Q8×C13)⋊2C22, C4.5(C22×D13), (C4×D13).2C22, (D4×C13).3C22, SmallGroup(416,135)

Series: Derived Chief Lower central Upper central

C1C52 — Q8⋊D26
C1C13C26C52C4×D13D4×D13 — Q8⋊D26
C13C26C52 — Q8⋊D26
C1C2C4SD16

Generators and relations for Q8⋊D26
 G = < a,b,c,d | a4=c26=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 632 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×6], C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C13, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, D13 [×3], C26, C26, C8⋊C22, Dic13, C52, C52, D26, D26 [×4], C2×C26, C132C8, C104, C4×D13, C4×D13, D52 [×2], D52, C13⋊D4, D4×C13, Q8×C13, C22×D13, C8⋊D13, D104, D4⋊D13, Q8⋊D13, C13×SD16, D4×D13, D52⋊C2, Q8⋊D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8⋊C22, D26 [×3], C22×D13, D4×D13, Q8⋊D26

Smallest permutation representation of Q8⋊D26
On 104 points
Generators in S104
(1 28 16 41)(2 42 17 29)(3 30 18 43)(4 44 19 31)(5 32 20 45)(6 46 21 33)(7 34 22 47)(8 48 23 35)(9 36 24 49)(10 50 25 37)(11 38 26 51)(12 52 14 39)(13 40 15 27)(53 66 82 95)(54 96 83 67)(55 68 84 97)(56 98 85 69)(57 70 86 99)(58 100 87 71)(59 72 88 101)(60 102 89 73)(61 74 90 103)(62 104 91 75)(63 76 92 79)(64 80 93 77)(65 78 94 81)
(1 55 16 84)(2 69 17 98)(3 57 18 86)(4 71 19 100)(5 59 20 88)(6 73 21 102)(7 61 22 90)(8 75 23 104)(9 63 24 92)(10 77 25 80)(11 65 26 94)(12 53 14 82)(13 67 15 96)(27 54 40 83)(28 97 41 68)(29 56 42 85)(30 99 43 70)(31 58 44 87)(32 101 45 72)(33 60 46 89)(34 103 47 74)(35 62 48 91)(36 79 49 76)(37 64 50 93)(38 81 51 78)(39 66 52 95)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(27 33)(28 32)(29 31)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(75 78)(76 77)(79 80)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E4A4B4C8A8B13A···13F26A···26F26G···26L52A···52F52G···52L104A···104L
order1222224448813···1326···2626···2652···5252···52104···104
size11426525224264522···22···28···84···48···84···4

53 irreducible representations

dim11111111222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D13D26D26D26C8⋊C22D4×D13Q8⋊D26
kernelQ8⋊D26C8⋊D13D104D4⋊D13Q8⋊D13C13×SD16D4×D13D52⋊C2Dic13D26SD16C8D4Q8C13C2C1
# reps111111111166661612

Matrix representation of Q8⋊D26 in GL4(𝔽313) generated by

1010879
01306161
2911083120
244980312
,
0018644
008661
2937600
17713400
,
208187184215
179182217249
00304126
00216245
,
221110249113
1829223281
00861
00117227
G:=sub<GL(4,GF(313))| [1,0,291,244,0,1,108,98,108,306,312,0,79,161,0,312],[0,0,293,177,0,0,76,134,186,86,0,0,44,61,0,0],[208,179,0,0,187,182,0,0,184,217,304,216,215,249,126,245],[221,182,0,0,110,92,0,0,249,232,86,117,113,81,1,227] >;

Q8⋊D26 in GAP, Magma, Sage, TeX

Q_8\rtimes D_{26}
% in TeX

G:=Group("Q8:D26");
// GroupNames label

G:=SmallGroup(416,135);
// by ID

G=gap.SmallGroup(416,135);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,116,86,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^26=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

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