metabelian, supersoluble, monomial
Aliases: C52⋊7D4, C102⋊3C2, C10.16D10, (C2×C10)⋊2D5, C22⋊(C5⋊D5), C5⋊3(C5⋊D4), C52⋊6C4⋊3C2, (C5×C10).15C22, (C2×C5⋊D5)⋊3C2, C2.5(C2×C5⋊D5), SmallGroup(200,36)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2×C10, C52, C5⋊D4, C5⋊D5, C5×C10, C5×C10, C52⋊6C4, C2×C5⋊D5, C102, C52⋊7D4
Quotients: C1, C2, C22, D4, D5, D10, C5⋊D4, C5⋊D5, C2×C5⋊D5, 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 D5×C5⋊D4 C20.50D10 D4×C5⋊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 | 2A | 2B | 2C | 4 | 5A | ··· | 5L | 10A | ··· | 10AJ |
| order | 1 | 2 | 2 | 2 | 4 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 1 | 2 | 50 | 50 | 2 | ··· | 2 | 2 | ··· | 2 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | C5⋊D4 |
| kernel | C52⋊7D4 | C52⋊6C4 | C2×C5⋊D5 | C102 | C52 | C2×C10 | C10 | C5 |
| # reps | 1 | 1 | 1 | 1 | 1 | 12 | 12 | 24 |
Matrix representation of C52⋊7D4 ►in GL4(𝔽41) generated by
| 40 | 34 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 38 | 3 |
| 0 | 0 | 24 | 3 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 40 | 34 |
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