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G = D524C4order 416 = 25·13

1st semidirect product of D52 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D524C4, C52.33D4, C4.17D52, C423D13, Dic264C4, C133C4≀C2, (C4×C52)⋊6C2, C4.6(C4×D13), C52.37(C2×C4), (C2×C26).26D4, (C2×C4).66D26, C52.4C41C2, D525C2.1C2, (C2×C52).96C22, C2.3(D26⋊C4), C26.12(C22⋊C4), C22.7(C13⋊D4), SmallGroup(416,12)

Series: Derived Chief Lower central Upper central

C1C52 — D524C4
C1C13C26C2×C26C2×C52D525C2 — D524C4
C13C26C52 — D524C4
C1C4C2×C4C42

Generators and relations for D524C4
 G = < a,b,c | a52=b2=c4=1, bab=a-1, ac=ca, cbc-1=a39b >

2C2
52C2
2C4
2C4
26C22
26C4
2C26
4D13
2C2×C4
13D4
13Q8
26C8
26D4
26C2×C4
2C52
2C52
2D26
2Dic13
13M4(2)
13C4○D4
2C13⋊D4
2C4×D13
2C132C8
2C2×C52
13C4≀C2

Smallest permutation representation of D524C4
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 82)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 104)(32 103)(33 102)(34 101)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(53 66 79 92)(54 67 80 93)(55 68 81 94)(56 69 82 95)(57 70 83 96)(58 71 84 97)(59 72 85 98)(60 73 86 99)(61 74 87 100)(62 75 88 101)(63 76 89 102)(64 77 90 103)(65 78 91 104)

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

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

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

110 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H8A8B13A···13F26A···26R52A···52BT
order1222444···448813···1326···2652···52
size11252112···25252522···22···22···2

110 irreducible representations

dim111111222222222
type+++++++++
imageC1C2C2C2C4C4D4D4D13C4≀C2D26C4×D13D52C13⋊D4D524C4
kernelD524C4C52.4C4C4×C52D525C2Dic26D52C52C2×C26C42C13C2×C4C4C4C22C1
# reps1111221164612121248

Matrix representation of D524C4 in GL2(𝔽313) generated by

2640
0198
,
0198
2640
,
3120
025
G:=sub<GL(2,GF(313))| [264,0,0,198],[0,264,198,0],[312,0,0,25] >;

D524C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes_4C_4
% in TeX

G:=Group("D52:4C4");
// GroupNames label

G:=SmallGroup(416,12);
// by ID

G=gap.SmallGroup(416,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,362,579,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^39*b>;
// generators/relations

Export

Subgroup lattice of D524C4 in TeX

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