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G = D525C2order 208 = 24·13

The semidirect product of D52 and C2 acting through Inn(D52)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D525C2, C4.16D26, Dic265C2, C26.4C23, C22.2D26, C52.16C22, D26.1C22, Dic13.2C22, (C2×C52)⋊4C2, (C2×C4)⋊3D13, (C4×D13)⋊4C2, C131(C4○D4), C13⋊D43C2, C2.5(C22×D13), (C2×C26).11C22, SmallGroup(208,38)

Series: Derived Chief Lower central Upper central

C1C26 — D525C2
C1C13C26D26C4×D13 — D525C2
C13C26 — D525C2
C1C4C2×C4

Generators and relations for D525C2
 G = < a,b,c | a52=b2=c2=1, bab=a-1, ac=ca, cbc=a26b >

2C2
26C2
26C2
13C4
13C4
13C22
13C22
2C26
2D13
2D13
13C2×C4
13D4
13D4
13D4
13C2×C4
13Q8
13C4○D4

Smallest permutation representation of D525C2
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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 59)(54 58)(55 57)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)

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

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

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

D525C2 is a maximal subgroup of
D524C4  D527C4  D52.3C4  D1047C2  D52.2C4  C8⋊D26  C8.D26  D526C22  Q8.D26  C52.C23  D46D26  Q8.10D26  C4○D4×D13  D48D26  D4.10D26
D525C2 is a maximal quotient of
C4×Dic26  C52.6Q8  C42⋊D13  C4×D52  C4.D52  C422D13  C23.D26  D26.12D4  D26⋊D4  C23.6D26  Dic13.Q8  D26.13D4  D26⋊Q8  C4⋊C4⋊D13  C52.48D4  C23.21D26  C4×C13⋊D4  C23.23D26  C527D4

58 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E13A···13F26A···26R52A···52X
order122224444413···1326···2652···52
size112262611226262···22···22···2

58 irreducible representations

dim11111122222
type+++++++++
imageC1C2C2C2C2C2C4○D4D13D26D26D525C2
kernelD525C2Dic26C4×D13D52C13⋊D4C2×C52C13C2×C4C4C22C1
# reps1121212612624

Matrix representation of D525C2 in GL2(𝔽53) generated by

3340
2744
,
952
2744
,
2818
3625
G:=sub<GL(2,GF(53))| [33,27,40,44],[9,27,52,44],[28,36,18,25] >;

D525C2 in GAP, Magma, Sage, TeX

D_{52}\rtimes_5C_2
% in TeX

G:=Group("D52:5C2");
// GroupNames label

G:=SmallGroup(208,38);
// by ID

G=gap.SmallGroup(208,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,46,182,4804]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^26*b>;
// generators/relations

Export

Subgroup lattice of D525C2 in TeX

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