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G = C52:7D4order 200 = 23·52

2nd semidirect product of C52 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial

Aliases: C52:7D4, C102:3C2, C10.16D10, (C2xC10):2D5, C22:(C5:D5), C5:3(C5:D4), C52:6C4:3C2, (C5xC10).15C22, (C2xC5:D5):3C2, C2.5(C2xC5:D5), SmallGroup(200,36)

Series: Derived Chief Lower central Upper central

C1C5xC10 — C52:7D4
C1C5C52C5xC10C2xC5:D5 — C52:7D4
C52C5xC10 — C52:7D4
C1C2C22

Generators and relations for C52:7D4
 G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 320 in 64 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C5, D4, D5, C10, C10, Dic5, D10, C2xC10, C52, C5:D4, C5:D5, C5xC10, C5xC10, C52:6C4, C2xC5:D5, C102, C52:7D4
Quotients: C1, C2, C22, D4, D5, D10, C5:D4, C5:D5, C2xC5:D5, C52:7D4

Smallest permutation representation of C52:7D4
On 100 points
Generators in S100
(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)
(1 44 38 33 28)(2 45 39 34 29)(3 41 40 35 30)(4 42 36 31 26)(5 43 37 32 27)(6 100 22 16 11)(7 96 23 17 12)(8 97 24 18 13)(9 98 25 19 14)(10 99 21 20 15)(46 67 61 56 51)(47 68 62 57 52)(48 69 63 58 53)(49 70 64 59 54)(50 66 65 60 55)(71 92 86 81 76)(72 93 87 82 77)(73 94 88 83 78)(74 95 89 84 79)(75 91 90 85 80)
(1 73 48 98)(2 72 49 97)(3 71 50 96)(4 75 46 100)(5 74 47 99)(6 42 80 67)(7 41 76 66)(8 45 77 70)(9 44 78 69)(10 43 79 68)(11 36 85 61)(12 40 81 65)(13 39 82 64)(14 38 83 63)(15 37 84 62)(16 31 90 56)(17 35 86 60)(18 34 87 59)(19 33 88 58)(20 32 89 57)(21 27 95 52)(22 26 91 51)(23 30 92 55)(24 29 93 54)(25 28 94 53)
(2 5)(3 4)(6 92)(7 91)(8 95)(9 94)(10 93)(11 86)(12 90)(13 89)(14 88)(15 87)(16 81)(17 85)(18 84)(19 83)(20 82)(21 77)(22 76)(23 80)(24 79)(25 78)(26 41)(27 45)(28 44)(29 43)(30 42)(31 40)(32 39)(33 38)(34 37)(35 36)(46 50)(47 49)(51 66)(52 70)(53 69)(54 68)(55 67)(56 65)(57 64)(58 63)(59 62)(60 61)(71 100)(72 99)(73 98)(74 97)(75 96)

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

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

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

C52:7D4 is a maximal subgroup of   Dic5.D10  D5xC5:D4  C20.50D10  D4xC5:D5  C20.D10
C52:7D4 is a maximal quotient of   C102.22C22  C10.11D20  C52:7D8  C52:8SD16  C52:10SD16  C52:7Q16  C102:11C4

53 conjugacy classes

class 1 2A2B2C 4 5A···5L10A···10AJ
order122245···510···10
size11250502···22···2

53 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D5D10C5:D4
kernelC52:7D4C52:6C4C2xC5:D5C102C52C2xC10C10C5
# reps11111121224

Matrix representation of C52:7D4 in GL4(F41) generated by

403400
7700
0001
004034
,
0100
403400
0001
004034
,
174000
32400
00383
00243
,
1000
344000
0077
004034
G:=sub<GL(4,GF(41))| [40,7,0,0,34,7,0,0,0,0,0,40,0,0,1,34],[0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,34],[17,3,0,0,40,24,0,0,0,0,38,24,0,0,3,3],[1,34,0,0,0,40,0,0,0,0,7,40,0,0,7,34] >;

C52:7D4 in GAP, Magma, Sage, TeX

C_5^2\rtimes_7D_4
% in TeX

G:=Group("C5^2:7D4");
// GroupNames label

G:=SmallGroup(200,36);
// by ID

G=gap.SmallGroup(200,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,61,643,4004]);
// Polycyclic

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

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