Copied to
clipboard

G = D26.4D4order 416 = 25·13

4th non-split extension by D26 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D26.4D4, C23⋊(C13⋊C4), C132(C23⋊C4), (C22×C26)⋊3C4, D13.D42C2, (C2×Dic13)⋊2C4, C26.9(C22⋊C4), C2.10(D13.D4), (C22×D13).15C22, C22.4(C2×C13⋊C4), (C2×C26).10(C2×C4), (C2×C13⋊D4).7C2, SmallGroup(416,86)

Series: Derived Chief Lower central Upper central

C1C2×C26 — D26.4D4
C1C13C26D26C22×D13D13.D4 — D26.4D4
C13C26C2×C26 — D26.4D4
C1C2C22C23

Generators and relations for D26.4D4
 G = < a,b,c,d | a26=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a5, cbc-1=a17b, dbd-1=a4b, dcd-1=a-1bc-1 >

2C2
4C2
26C2
26C2
2C22
4C22
13C22
13C22
26C4
52C4
52C22
52C4
2D13
2C26
2D13
4C26
13C2×C4
13C23
26D4
26D4
26C2×C4
26C2×C4
2Dic13
2C2×C26
4C13⋊C4
4D26
4C13⋊C4
4C2×C26
13C2×D4
13C22⋊C4
13C22⋊C4
2C2×C13⋊C4
2C2×C13⋊C4
2C13⋊D4
2C13⋊D4
13C23⋊C4

Smallest permutation representation of D26.4D4
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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)
(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 40)(28 35 52 45)(29 30 51 50)(31 46 49 34)(32 41 48 39)(33 36 47 44)(37 42 43 38)(53 86 70 79)(54 81 69 84)(55 102 68 89)(56 97 67 94)(57 92 66 99)(58 87 65 104)(59 82 64 83)(60 103 63 88)(61 98 62 93)(71 100 78 91)(72 95 77 96)(73 90 76 101)(74 85 75 80)
(1 68 27 102)(2 63 52 81)(3 58 51 86)(4 53 50 91)(5 74 49 96)(6 69 48 101)(7 64 47 80)(8 59 46 85)(9 54 45 90)(10 75 44 95)(11 70 43 100)(12 65 42 79)(13 60 41 84)(14 55 40 89)(15 76 39 94)(16 71 38 99)(17 66 37 104)(18 61 36 83)(19 56 35 88)(20 77 34 93)(21 72 33 98)(22 67 32 103)(23 62 31 82)(24 57 30 87)(25 78 29 92)(26 73 28 97)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E4A···4E13A13B13C26A···26U
order1222224···413131326···26
size1124262652···524444···4

35 irreducible representations

dim11111244444
type++++++++
imageC1C2C2C4C4D4C23⋊C4C13⋊C4C2×C13⋊C4D13.D4D26.4D4
kernelD26.4D4D13.D4C2×C13⋊D4C2×Dic13C22×C26D26C13C23C22C2C1
# reps121222133612

Matrix representation of D26.4D4 in GL4(𝔽53) generated by

540420
33343433
204405
48142934
,
1734840
340936
47244350
1013366
,
1000
33281434
5392419
0010
,
1749829
18281824
32113224
48183929
G:=sub<GL(4,GF(53))| [5,33,20,48,40,34,4,14,4,34,40,29,20,33,5,34],[17,3,47,10,34,40,24,13,8,9,43,36,40,36,50,6],[1,33,5,0,0,28,39,0,0,14,24,1,0,34,19,0],[17,18,32,48,49,28,11,18,8,18,32,39,29,24,24,29] >;

D26.4D4 in GAP, Magma, Sage, TeX

D_{26}._4D_4
% in TeX

G:=Group("D26.4D4");
// GroupNames label

G:=SmallGroup(416,86);
// by ID

G=gap.SmallGroup(416,86);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,579,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^17*b,d*b*d^-1=a^4*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

Export

Subgroup lattice of D26.4D4 in TeX

׿
×
𝔽