metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.11D52, C52.46D4, M4(2)⋊3D13, (C2×C4).1D26, (C2×D52).7C2, C52.4C4⋊2C2, C13⋊2(C4.D4), C22.4(C4×D13), C4.21(C13⋊D4), (C13×M4(2))⋊7C2, (C2×C52).13C22, (C22×D13).1C4, C2.9(D26⋊C4), C26.19(C22⋊C4), (C2×C26).22(C2×C4), SmallGroup(416,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊D13
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >
2C2
52C2
52C2
26C22
26C22
52C22
52C22
2C26
4D13
4D13
2C8
13C23
13C23
26C8
26D4
26D4
2D26
2D26
4D26
4D26
13M4(2)
13C2×D4
2C104
2D52
2D52
13C4.D4
Smallest permutation representation of M4(2)⋊D13
►On 104 points
Generators in S104
(1 90 29 77 25 100 48 64)(2 91 30 78 26 101 49 65)(3 79 31 66 14 102 50 53)(4 80 32 67 15 103 51 54)(5 81 33 68 16 104 52 55)(6 82 34 69 17 92 40 56)(7 83 35 70 18 93 41 57)(8 84 36 71 19 94 42 58)(9 85 37 72 20 95 43 59)(10 86 38 73 21 96 44 60)(11 87 39 74 22 97 45 61)(12 88 27 75 23 98 46 62)(13 89 28 76 24 99 47 63)
(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(79 102)(80 103)(81 104)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(91 101)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 52)(38 51)(39 50)(53 61)(54 60)(55 59)(56 58)(62 65)(63 64)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)
G:=sub<Sym(104)| (1,90,29,77,25,100,48,64)(2,91,30,78,26,101,49,65)(3,79,31,66,14,102,50,53)(4,80,32,67,15,103,51,54)(5,81,33,68,16,104,52,55)(6,82,34,69,17,92,40,56)(7,83,35,70,18,93,41,57)(8,84,36,71,19,94,42,58)(9,85,37,72,20,95,43,59)(10,86,38,73,21,96,44,60)(11,87,39,74,22,97,45,61)(12,88,27,75,23,98,46,62)(13,89,28,76,24,99,47,63), (53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(79,102)(80,103)(81,104)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(91,101), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,52)(38,51)(39,50)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)>;
G:=Group( (1,90,29,77,25,100,48,64)(2,91,30,78,26,101,49,65)(3,79,31,66,14,102,50,53)(4,80,32,67,15,103,51,54)(5,81,33,68,16,104,52,55)(6,82,34,69,17,92,40,56)(7,83,35,70,18,93,41,57)(8,84,36,71,19,94,42,58)(9,85,37,72,20,95,43,59)(10,86,38,73,21,96,44,60)(11,87,39,74,22,97,45,61)(12,88,27,75,23,98,46,62)(13,89,28,76,24,99,47,63), (53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(79,102)(80,103)(81,104)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(91,101), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,52)(38,51)(39,50)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98) );
G=PermutationGroup([(1,90,29,77,25,100,48,64),(2,91,30,78,26,101,49,65),(3,79,31,66,14,102,50,53),(4,80,32,67,15,103,51,54),(5,81,33,68,16,104,52,55),(6,82,34,69,17,92,40,56),(7,83,35,70,18,93,41,57),(8,84,36,71,19,94,42,58),(9,85,37,72,20,95,43,59),(10,86,38,73,21,96,44,60),(11,87,39,74,22,97,45,61),(12,88,27,75,23,98,46,62),(13,89,28,76,24,99,47,63)], [(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78),(79,102),(80,103),(81,104),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(91,101)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,52),(38,51),(39,50),(53,61),(54,60),(55,59),(56,58),(62,65),(63,64),(66,74),(67,73),(68,72),(69,71),(75,78),(76,77),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(91,98)])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 52 | 52 | 2 | 2 | 4 | 4 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D13 | D26 | D52 | C13⋊D4 | C4×D13 | C4.D4 | M4(2)⋊D13 |
| kernel | M4(2)⋊D13 | C52.4C4 | C13×M4(2) | C2×D52 | C22×D13 | C52 | M4(2) | C2×C4 | C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 6 | 12 | 12 | 12 | 1 | 12 |
Matrix representation of M4(2)⋊D13 ►in GL4(𝔽313) generated by
| 155 | 296 | 311 | 0 |
| 90 | 221 | 0 | 311 |
| 59 | 137 | 158 | 17 |
| 222 | 101 | 223 | 92 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 155 | 296 | 312 | 0 |
| 90 | 221 | 0 | 312 |
| 6 | 1 | 0 | 0 |
| 277 | 255 | 0 | 0 |
| 261 | 198 | 6 | 1 |
| 127 | 52 | 277 | 255 |
| 235 | 37 | 0 | 0 |
| 64 | 78 | 0 | 0 |
| 295 | 270 | 235 | 37 |
| 100 | 18 | 64 | 78 |
G:=sub<GL(4,GF(313))| [155,90,59,222,296,221,137,101,311,0,158,223,0,311,17,92],[1,0,155,90,0,1,296,221,0,0,312,0,0,0,0,312],[6,277,261,127,1,255,198,52,0,0,6,277,0,0,1,255],[235,64,295,100,37,78,270,18,0,0,235,64,0,0,37,78] >;
M4(2)⋊D13 in GAP, Magma, Sage, TeX
M_{4(2})\rtimes D_{13} % in TeX
G:=Group("M4(2):D13"); // GroupNames label
G:=SmallGroup(416,30);
// by ID
G=gap.SmallGroup(416,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,362,86,297,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations