direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)×D13, C8⋊6D26, C104⋊6C22, C52.38C23, (C8×D13)⋊7C2, C8⋊D13⋊5C2, C52.33(C2×C4), (C4×D13).1C4, C4.15(C4×D13), (C2×C4).45D26, C13⋊5(C2×M4(2)), C52.4C4⋊5C2, D26.11(C2×C4), C22.7(C4×D13), C13⋊2C8⋊11C22, (C13×M4(2))⋊3C2, (C2×C52).25C22, C26.28(C22×C4), (C2×Dic13).7C4, (C22×D13).5C4, C4.38(C22×D13), Dic13.13(C2×C4), (C4×D13).38C22, (C2×C4×D13).4C2, C2.16(C2×C4×D13), (C2×C26).25(C2×C4), SmallGroup(416,127)
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, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 400 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C13, C2×C8, M4(2), M4(2), C22×C4, D13, D13, C26, C26, C2×M4(2), Dic13, C52, D26, D26, C2×C26, C13⋊2C8, C104, C4×D13, C2×Dic13, C2×C52, C22×D13, C8×D13, C8⋊D13, C52.4C4, C13×M4(2), C2×C4×D13, M4(2)×D13
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, D13, C2×M4(2), D26, C4×D13, C22×D13, C2×C4×D13, M4(2)×D13
►On 104 points
Generators in S104
(1 89 39 78 19 102 44 61)(2 90 27 66 20 103 45 62)(3 91 28 67 21 104 46 63)(4 79 29 68 22 92 47 64)(5 80 30 69 23 93 48 65)(6 81 31 70 24 94 49 53)(7 82 32 71 25 95 50 54)(8 83 33 72 26 96 51 55)(9 84 34 73 14 97 52 56)(10 85 35 74 15 98 40 57)(11 86 36 75 16 99 41 58)(12 87 37 76 17 100 42 59)(13 88 38 77 18 101 43 60)
(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 66)(63 67)(64 68)(65 69)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 23)(15 22)(16 21)(17 20)(18 19)(24 26)(27 37)(28 36)(29 35)(30 34)(31 33)(38 39)(40 47)(41 46)(42 45)(43 44)(48 52)(49 51)(53 55)(56 65)(57 64)(58 63)(59 62)(60 61)(66 76)(67 75)(68 74)(69 73)(70 72)(77 78)(79 85)(80 84)(81 83)(86 91)(87 90)(88 89)(92 98)(93 97)(94 96)(99 104)(100 103)(101 102)
G:=sub<Sym(104)| (1,89,39,78,19,102,44,61)(2,90,27,66,20,103,45,62)(3,91,28,67,21,104,46,63)(4,79,29,68,22,92,47,64)(5,80,30,69,23,93,48,65)(6,81,31,70,24,94,49,53)(7,82,32,71,25,95,50,54)(8,83,33,72,26,96,51,55)(9,84,34,73,14,97,52,56)(10,85,35,74,15,98,40,57)(11,86,36,75,16,99,41,58)(12,87,37,76,17,100,42,59)(13,88,38,77,18,101,43,60), (53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,66)(63,67)(64,68)(65,69)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)(27,37)(28,36)(29,35)(30,34)(31,33)(38,39)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,55)(56,65)(57,64)(58,63)(59,62)(60,61)(66,76)(67,75)(68,74)(69,73)(70,72)(77,78)(79,85)(80,84)(81,83)(86,91)(87,90)(88,89)(92,98)(93,97)(94,96)(99,104)(100,103)(101,102)>;
G:=Group( (1,89,39,78,19,102,44,61)(2,90,27,66,20,103,45,62)(3,91,28,67,21,104,46,63)(4,79,29,68,22,92,47,64)(5,80,30,69,23,93,48,65)(6,81,31,70,24,94,49,53)(7,82,32,71,25,95,50,54)(8,83,33,72,26,96,51,55)(9,84,34,73,14,97,52,56)(10,85,35,74,15,98,40,57)(11,86,36,75,16,99,41,58)(12,87,37,76,17,100,42,59)(13,88,38,77,18,101,43,60), (53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,66)(63,67)(64,68)(65,69)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)(27,37)(28,36)(29,35)(30,34)(31,33)(38,39)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,55)(56,65)(57,64)(58,63)(59,62)(60,61)(66,76)(67,75)(68,74)(69,73)(70,72)(77,78)(79,85)(80,84)(81,83)(86,91)(87,90)(88,89)(92,98)(93,97)(94,96)(99,104)(100,103)(101,102) );
G=PermutationGroup([[(1,89,39,78,19,102,44,61),(2,90,27,66,20,103,45,62),(3,91,28,67,21,104,46,63),(4,79,29,68,22,92,47,64),(5,80,30,69,23,93,48,65),(6,81,31,70,24,94,49,53),(7,82,32,71,25,95,50,54),(8,83,33,72,26,96,51,55),(9,84,34,73,14,97,52,56),(10,85,35,74,15,98,40,57),(11,86,36,75,16,99,41,58),(12,87,37,76,17,100,42,59),(13,88,38,77,18,101,43,60)], [(53,70),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,66),(63,67),(64,68),(65,69),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,23),(15,22),(16,21),(17,20),(18,19),(24,26),(27,37),(28,36),(29,35),(30,34),(31,33),(38,39),(40,47),(41,46),(42,45),(43,44),(48,52),(49,51),(53,55),(56,65),(57,64),(58,63),(59,62),(60,61),(66,76),(67,75),(68,74),(69,73),(70,72),(77,78),(79,85),(80,84),(81,83),(86,91),(87,90),(88,89),(92,98),(93,97),(94,96),(99,104),(100,103),(101,102)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 13 | 13 | 26 | 1 | 1 | 2 | 13 | 13 | 26 | 2 | 2 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | D13 | D26 | D26 | C4×D13 | C4×D13 | M4(2)×D13 |
| kernel | M4(2)×D13 | C8×D13 | C8⋊D13 | C52.4C4 | C13×M4(2) | C2×C4×D13 | C4×D13 | C2×Dic13 | C22×D13 | D13 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 6 | 12 | 6 | 12 | 12 | 12 |
Matrix representation of M4(2)×D13 ►in GL4(𝔽313) generated by
| 288 | 264 | 0 | 0 |
| 140 | 25 | 0 | 0 |
| 0 | 0 | 288 | 0 |
| 0 | 0 | 0 | 288 |
| 1 | 0 | 0 | 0 |
| 197 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 312 | 24 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(313))| [288,140,0,0,264,25,0,0,0,0,288,0,0,0,0,288],[1,197,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,312,0,0,1,24],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
M4(2)×D13 in GAP, Magma, Sage, TeX
M_4(2)\times D_{13} % in TeX
G:=Group("M4(2)xD13"); // GroupNames label
G:=SmallGroup(416,127);
// by ID
G=gap.SmallGroup(416,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,50,69,13829]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations