metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D13, C26.11D4, C22⋊Dic13, C22.7D26, (C2×C26)⋊4C4, C13⋊3(C22⋊C4), C26.16(C2×C4), (C2×Dic13)⋊2C2, C2.3(C13⋊D4), (C2×C26).7C22, (C22×C26).2C2, C2.5(C2×Dic13), SmallGroup(208,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D13
G = < a,b,c,d,e | a2=b2=c2=d13=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Smallest permutation representation of C23.D13
►On 104 points
Generators in S104
(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 36)(2 37)(3 38)(4 39)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 40)(26 41)(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 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(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 91 36 65)(2 90 37 64)(3 89 38 63)(4 88 39 62)(5 87 27 61)(6 86 28 60)(7 85 29 59)(8 84 30 58)(9 83 31 57)(10 82 32 56)(11 81 33 55)(12 80 34 54)(13 79 35 53)(14 98 42 72)(15 97 43 71)(16 96 44 70)(17 95 45 69)(18 94 46 68)(19 93 47 67)(20 92 48 66)(21 104 49 78)(22 103 50 77)(23 102 51 76)(24 101 52 75)(25 100 40 74)(26 99 41 73)
G:=sub<Sym(104)| (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), (1,36)(2,37)(3,38)(4,39)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,40)(26,41)(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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(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), (1,91,36,65)(2,90,37,64)(3,89,38,63)(4,88,39,62)(5,87,27,61)(6,86,28,60)(7,85,29,59)(8,84,30,58)(9,83,31,57)(10,82,32,56)(11,81,33,55)(12,80,34,54)(13,79,35,53)(14,98,42,72)(15,97,43,71)(16,96,44,70)(17,95,45,69)(18,94,46,68)(19,93,47,67)(20,92,48,66)(21,104,49,78)(22,103,50,77)(23,102,51,76)(24,101,52,75)(25,100,40,74)(26,99,41,73)>;
G:=Group( (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), (1,36)(2,37)(3,38)(4,39)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,40)(26,41)(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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(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), (1,91,36,65)(2,90,37,64)(3,89,38,63)(4,88,39,62)(5,87,27,61)(6,86,28,60)(7,85,29,59)(8,84,30,58)(9,83,31,57)(10,82,32,56)(11,81,33,55)(12,80,34,54)(13,79,35,53)(14,98,42,72)(15,97,43,71)(16,96,44,70)(17,95,45,69)(18,94,46,68)(19,93,47,67)(20,92,48,66)(21,104,49,78)(22,103,50,77)(23,102,51,76)(24,101,52,75)(25,100,40,74)(26,99,41,73) );
G=PermutationGroup([[(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)], [(1,36),(2,37),(3,38),(4,39),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,40),(26,41),(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)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(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)], [(1,91,36,65),(2,90,37,64),(3,89,38,63),(4,88,39,62),(5,87,27,61),(6,86,28,60),(7,85,29,59),(8,84,30,58),(9,83,31,57),(10,82,32,56),(11,81,33,55),(12,80,34,54),(13,79,35,53),(14,98,42,72),(15,97,43,71),(16,96,44,70),(17,95,45,69),(18,94,46,68),(19,93,47,67),(20,92,48,66),(21,104,49,78),(22,103,50,77),(23,102,51,76),(24,101,52,75),(25,100,40,74),(26,99,41,73)]])
C23.D13 is a maximal subgroup of
C22.2D52 C23⋊Dic13 C23.11D26 C22⋊Dic26 C23.D26 C22⋊C4×D13 D26.12D4 C23.6D26 C52.48D4 C23.21D26 C4×C13⋊D4 C23.23D26 D4×Dic13 C23.18D26 C52.17D4 C23⋊D26 C52⋊2D4 Dic13⋊D4 C24⋊D13
C23.D13 is a maximal quotient of
C52.55D4 C26.10C42 D4⋊Dic13 C52.D4 C23⋊Dic13 Q8⋊Dic13 C52.10D4 C52.56D4
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 13A | ··· | 13F | 26A | ··· | 26AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C4 | D4 | D13 | Dic13 | D26 | C13⋊D4 |
| kernel | C23.D13 | C2×Dic13 | C22×C26 | C2×C26 | C26 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 6 | 12 | 6 | 24 |
Matrix representation of C23.D13 ►in GL4(𝔽53) generated by
| 1 | 0 | 0 | 0 |
| 23 | 52 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 37 |
| 0 | 0 | 0 | 24 |
| 47 | 42 | 0 | 0 |
| 13 | 6 | 0 | 0 |
| 0 | 0 | 49 | 21 |
| 0 | 0 | 22 | 4 |
G:=sub<GL(4,GF(53))| [1,23,0,0,0,52,0,0,0,0,1,0,0,0,10,52],[52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,42,0,0,0,37,24],[47,13,0,0,42,6,0,0,0,0,49,22,0,0,21,4] >;
C23.D13 in GAP, Magma, Sage, TeX
C_2^3.D_{13} % in TeX
G:=Group("C2^3.D13"); // GroupNames label
G:=SmallGroup(208,19);
// by ID
G=gap.SmallGroup(208,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,4804]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^13=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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