metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.6D26, C52.15D4, D52⋊6C22, C52.12C23, Dic26⋊5C22, D4⋊D13⋊5C2, (D4×C26)⋊2C2, (C2×D4)⋊2D13, C13⋊4(C8⋊C22), D4.D13⋊5C2, (C2×C4).17D26, C26.45(C2×D4), (C2×C26).39D4, D52⋊5C2⋊3C2, C52.4C4⋊6C2, C13⋊2C8⋊3C22, C4.16(C13⋊D4), (C2×C52).30C22, (D4×C13).6C22, C4.12(C22×D13), C22.10(C13⋊D4), C2.9(C2×C13⋊D4), SmallGroup(416,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊6C22
G = < a,b,c,d | a52=b2=c2=d2=1, bab=a-1, ac=ca, dad=a27, cbc=a26b, dbd=a39b, cd=dc >
Subgroups: 392 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, C52, D26, C2×C26, C2×C26, C13⋊2C8, Dic26, C4×D13, D52, C13⋊D4, C2×C52, D4×C13, D4×C13, C22×C26, C52.4C4, D4⋊D13, D4.D13, D52⋊5C2, D4×C26, D52⋊6C22
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C13⋊D4, C22×D13, C2×C13⋊D4, D52⋊6C22
►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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 104)(13 103)(14 102)(15 101)(16 100)(17 99)(18 98)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 90)(27 89)(28 88)(29 87)(30 86)(31 85)(32 84)(33 83)(34 82)(35 81)(36 80)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)
(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
(1 40)(2 15)(3 42)(4 17)(5 44)(6 19)(7 46)(8 21)(9 48)(10 23)(11 50)(12 25)(13 52)(14 27)(16 29)(18 31)(20 33)(22 35)(24 37)(26 39)(28 41)(30 43)(32 45)(34 47)(36 49)(38 51)(53 79)(55 81)(57 83)(59 85)(61 87)(63 89)(65 91)(67 93)(69 95)(71 97)(73 99)(75 101)(77 103)
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,104)(13,103)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,90)(27,89)(28,88)(29,87)(30,86)(31,85)(32,84)(33,83)(34,82)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64), (53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104), (1,40)(2,15)(3,42)(4,17)(5,44)(6,19)(7,46)(8,21)(9,48)(10,23)(11,50)(12,25)(13,52)(14,27)(16,29)(18,31)(20,33)(22,35)(24,37)(26,39)(28,41)(30,43)(32,45)(34,47)(36,49)(38,51)(53,79)(55,81)(57,83)(59,85)(61,87)(63,89)(65,91)(67,93)(69,95)(71,97)(73,99)(75,101)(77,103)>;
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,104)(13,103)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,90)(27,89)(28,88)(29,87)(30,86)(31,85)(32,84)(33,83)(34,82)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64), (53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104), (1,40)(2,15)(3,42)(4,17)(5,44)(6,19)(7,46)(8,21)(9,48)(10,23)(11,50)(12,25)(13,52)(14,27)(16,29)(18,31)(20,33)(22,35)(24,37)(26,39)(28,41)(30,43)(32,45)(34,47)(36,49)(38,51)(53,79)(55,81)(57,83)(59,85)(61,87)(63,89)(65,91)(67,93)(69,95)(71,97)(73,99)(75,101)(77,103) );
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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,104),(13,103),(14,102),(15,101),(16,100),(17,99),(18,98),(19,97),(20,96),(21,95),(22,94),(23,93),(24,92),(25,91),(26,90),(27,89),(28,88),(29,87),(30,86),(31,85),(32,84),(33,83),(34,82),(35,81),(36,80),(37,79),(38,78),(39,77),(40,76),(41,75),(42,74),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64)], [(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104)], [(1,40),(2,15),(3,42),(4,17),(5,44),(6,19),(7,46),(8,21),(9,48),(10,23),(11,50),(12,25),(13,52),(14,27),(16,29),(18,31),(20,33),(22,35),(24,37),(26,39),(28,41),(30,43),(32,45),(34,47),(36,49),(38,51),(53,79),(55,81),(57,83),(59,85),(61,87),(63,89),(65,91),(67,93),(69,95),(71,97),(73,99),(75,101),(77,103)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 4 | 4 | 52 | 2 | 2 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | C13⋊D4 | C13⋊D4 | C8⋊C22 | D52⋊6C22 |
| kernel | D52⋊6C22 | C52.4C4 | D4⋊D13 | D4.D13 | D52⋊5C2 | D4×C26 | C52 | C2×C26 | C2×D4 | C2×C4 | D4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 1 | 12 |
Matrix representation of D52⋊6C22 ►in GL4(𝔽313) generated by
| 58 | 140 | 0 | 0 |
| 256 | 255 | 0 | 0 |
| 221 | 99 | 286 | 175 |
| 290 | 229 | 65 | 27 |
| 221 | 99 | 286 | 175 |
| 56 | 35 | 237 | 84 |
| 58 | 140 | 0 | 0 |
| 165 | 61 | 154 | 57 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 312 | 0 |
| 103 | 260 | 0 | 312 |
| 1 | 24 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 103 | 297 | 299 | 312 |
G:=sub<GL(4,GF(313))| [58,256,221,290,140,255,99,229,0,0,286,65,0,0,175,27],[221,56,58,165,99,35,140,61,286,237,0,154,175,84,0,57],[1,0,0,103,0,1,0,260,0,0,312,0,0,0,0,312],[1,0,0,103,24,312,0,297,0,0,1,299,0,0,0,312] >;
D52⋊6C22 in GAP, Magma, Sage, TeX
D_{52}\rtimes_6C_2^2 % in TeX
G:=Group("D52:6C2^2"); // GroupNames label
G:=SmallGroup(416,153);
// by ID
G=gap.SmallGroup(416,153);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,579,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^52=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^27,c*b*c=a^26*b,d*b*d=a^39*b,c*d=d*c>;
// generators/relations