metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D26⋊5D4, C23⋊1D26, (C2×C4)⋊2D26, (C2×C26)⋊2D4, C13⋊2C22≀C2, (D4×C26)⋊8C2, (C2×D4)⋊3D13, (C2×C52)⋊7C22, C26.49(C2×D4), C2.25(D4×D13), (C23×D13)⋊2C2, D26⋊C4⋊14C2, C22⋊2(C13⋊D4), (C2×C26).52C23, (C22×C26)⋊3C22, C23.D13⋊10C2, (C2×Dic13)⋊2C22, C22.59(C22×D13), (C22×D13).28C22, (C2×C13⋊D4)⋊4C2, C2.13(C2×C13⋊D4), SmallGroup(416,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊D26
G = < a,b,c,d,e | a2=b2=c2=d26=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1064 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C2×D4, C2×D4, C24, D13, C26, C26, C26, C22≀C2, Dic13, C52, D26, D26, C2×C26, C2×C26, C2×C26, C2×Dic13, C13⋊D4, C2×C52, D4×C13, C22×D13, C22×D13, C22×C26, D26⋊C4, C23.D13, C2×C13⋊D4, D4×C26, C23×D13, C23⋊D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C22≀C2, D26, C13⋊D4, C22×D13, D4×D13, C2×C13⋊D4, C23⋊D26
►On 104 points
Generators in S104
(1 72)(2 60)(3 74)(4 62)(5 76)(6 64)(7 78)(8 66)(9 54)(10 68)(11 56)(12 70)(13 58)(14 67)(15 55)(16 69)(17 57)(18 71)(19 59)(20 73)(21 61)(22 75)(23 63)(24 77)(25 65)(26 53)(27 85)(28 99)(29 87)(30 101)(31 89)(32 103)(33 91)(34 79)(35 93)(36 81)(37 95)(38 83)(39 97)(40 96)(41 84)(42 98)(43 86)(44 100)(45 88)(46 102)(47 90)(48 104)(49 92)(50 80)(51 94)(52 82)
(1 38)(2 39)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 48)(15 49)(16 50)(17 51)(18 52)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 14)(10 15)(11 16)(12 17)(13 18)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 40)(39 41)(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 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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)
G:=sub<Sym(104)| (1,72)(2,60)(3,74)(4,62)(5,76)(6,64)(7,78)(8,66)(9,54)(10,68)(11,56)(12,70)(13,58)(14,67)(15,55)(16,69)(17,57)(18,71)(19,59)(20,73)(21,61)(22,75)(23,63)(24,77)(25,65)(26,53)(27,85)(28,99)(29,87)(30,101)(31,89)(32,103)(33,91)(34,79)(35,93)(36,81)(37,95)(38,83)(39,97)(40,96)(41,84)(42,98)(43,86)(44,100)(45,88)(46,102)(47,90)(48,104)(49,92)(50,80)(51,94)(52,82), (1,38)(2,39)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,48)(15,49)(16,50)(17,51)(18,52)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,14)(10,15)(11,16)(12,17)(13,18)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,40)(39,41)(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)(79,92)(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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)>;
G:=Group( (1,72)(2,60)(3,74)(4,62)(5,76)(6,64)(7,78)(8,66)(9,54)(10,68)(11,56)(12,70)(13,58)(14,67)(15,55)(16,69)(17,57)(18,71)(19,59)(20,73)(21,61)(22,75)(23,63)(24,77)(25,65)(26,53)(27,85)(28,99)(29,87)(30,101)(31,89)(32,103)(33,91)(34,79)(35,93)(36,81)(37,95)(38,83)(39,97)(40,96)(41,84)(42,98)(43,86)(44,100)(45,88)(46,102)(47,90)(48,104)(49,92)(50,80)(51,94)(52,82), (1,38)(2,39)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,48)(15,49)(16,50)(17,51)(18,52)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,14)(10,15)(11,16)(12,17)(13,18)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,40)(39,41)(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)(79,92)(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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91) );
G=PermutationGroup([[(1,72),(2,60),(3,74),(4,62),(5,76),(6,64),(7,78),(8,66),(9,54),(10,68),(11,56),(12,70),(13,58),(14,67),(15,55),(16,69),(17,57),(18,71),(19,59),(20,73),(21,61),(22,75),(23,63),(24,77),(25,65),(26,53),(27,85),(28,99),(29,87),(30,101),(31,89),(32,103),(33,91),(34,79),(35,93),(36,81),(37,95),(38,83),(39,97),(40,96),(41,84),(42,98),(43,86),(44,100),(45,88),(46,102),(47,90),(48,104),(49,92),(50,80),(51,94),(52,82)], [(1,38),(2,39),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,48),(15,49),(16,50),(17,51),(18,52),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,97),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,14),(10,15),(11,16),(12,17),(13,18),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,40),(39,41),(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),(79,92),(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(27,52),(28,51),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,104),(66,103),(67,102),(68,101),(69,100),(70,99),(71,98),(72,97),(73,96),(74,95),(75,94),(76,93),(77,92),(78,91)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 26 | 26 | 26 | 26 | 4 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | C13⋊D4 | D4×D13 |
| kernel | C23⋊D26 | D26⋊C4 | C23.D13 | C2×C13⋊D4 | D4×C26 | C23×D13 | D26 | C2×C26 | C2×D4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 6 | 6 | 12 | 24 | 12 |
Matrix representation of C23⋊D26 ►in GL4(𝔽53) generated by
| 48 | 17 | 0 | 0 |
| 36 | 5 | 0 | 0 |
| 0 | 0 | 52 | 12 |
| 0 | 0 | 0 | 1 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 51 | 51 | 0 | 0 |
| 2 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 52 |
| 12 | 15 | 0 | 0 |
| 47 | 41 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 44 | 1 |
G:=sub<GL(4,GF(53))| [48,36,0,0,17,5,0,0,0,0,52,0,0,0,12,1],[52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[51,2,0,0,51,28,0,0,0,0,1,9,0,0,0,52],[12,47,0,0,15,41,0,0,0,0,52,44,0,0,0,1] >;
C23⋊D26 in GAP, Magma, Sage, TeX
C_2^3\rtimes D_{26} % in TeX
G:=Group("C2^3:D26"); // GroupNames label
G:=SmallGroup(416,158);
// by ID
G=gap.SmallGroup(416,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,13829]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^26=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations