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G = D26.Q8order 416 = 25·13

3rd non-split extension by D26 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D26.3Q8, D26.10D4, C26.5C42, (C2×C52)⋊2C4, D13.(C4⋊C4), C26.7(C4⋊C4), D26.7(C2×C4), (C2×Dic13)⋊6C4, C13⋊(C2.C42), D13.(C22⋊C4), C2.3(C52⋊C4), C26.4(C22⋊C4), C2.2(D13.D4), (C22×D13).35C22, (C2×C13⋊C4)⋊C4, C2.5(C4×C13⋊C4), (C2×C4)⋊2(C13⋊C4), (C2×C4×D13).9C2, (C2×C26).9(C2×C4), (C22×C13⋊C4).1C2, C22.13(C2×C13⋊C4), SmallGroup(416,81)

Series: Derived Chief Lower central Upper central

C1C26 — D26.Q8
C1C13D13D26C22×D13C22×C13⋊C4 — D26.Q8
C13C26 — D26.Q8
C1C22C2×C4

Generators and relations for D26.Q8
 G = < a,b,c,d | a26=b2=c4=1, d2=a12bc2, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=a13c-1 >

Subgroups: 604 in 76 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×6], C2×C4, C2×C4 [×11], C23, C13, C22×C4 [×3], D13 [×4], C26 [×3], C2.C42, Dic13, C52, C13⋊C4 [×4], D26 [×2], D26 [×4], C2×C26, C4×D13 [×2], C2×Dic13, C2×C52, C2×C13⋊C4 [×4], C2×C13⋊C4 [×4], C22×D13, C2×C4×D13, C22×C13⋊C4 [×2], D26.Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C13⋊C4, C2×C13⋊C4, C4×C13⋊C4, C52⋊C4, D13.D4, D26.Q8

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C···4L13A13B13C26A···26I52A···52L
order12222222444···413131326···2652···52
size1111131313132226···264444···44···4

44 irreducible representations

dim1111112244444
type++++-+++
imageC1C2C2C4C4C4D4Q8C13⋊C4C2×C13⋊C4C4×C13⋊C4C52⋊C4D13.D4
kernelD26.Q8C2×C4×D13C22×C13⋊C4C2×Dic13C2×C52C2×C13⋊C4D26D26C2×C4C22C2C2C2
# reps1122283133666

Matrix representation of D26.Q8 in GL8(𝔽53)

520000000
052000000
00100000
00010000
000022282314
000039383839
000014232822
000031241638
,
10000000
01000000
00100000
00010000
000022282314
000038162431
000014362915
000052393139
,
3243000000
4421000000
0010400000
0020430000
000052000
000005200
000000520
000000052
,
10000000
1752000000
002300000
0015300000
00001000
000014362915
000031241638
00000010

G:=sub<GL(8,GF(53))| [52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,39,14,31,0,0,0,0,28,38,23,24,0,0,0,0,23,38,28,16,0,0,0,0,14,39,22,38],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,22,38,14,52,0,0,0,0,28,16,36,39,0,0,0,0,23,24,29,31,0,0,0,0,14,31,15,39],[32,44,0,0,0,0,0,0,43,21,0,0,0,0,0,0,0,0,10,20,0,0,0,0,0,0,40,43,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,52],[1,17,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,0,23,15,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,14,31,0,0,0,0,0,0,36,24,0,0,0,0,0,0,29,16,1,0,0,0,0,0,15,38,0] >;

D26.Q8 in GAP, Magma, Sage, TeX

D_{26}.Q_8
% in TeX

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

G:=SmallGroup(416,81);
// by ID

G=gap.SmallGroup(416,81);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^12*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^13*c^-1>;
// generators/relations

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