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G = C23⋊D26order 416 = 25·13

1st semidirect product of C23 and D26 acting via D26/C13=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D265D4, C231D26, (C2×C4)⋊2D26, (C2×C26)⋊2D4, C132C22≀C2, (D4×C26)⋊8C2, (C2×D4)⋊3D13, (C2×C52)⋊7C22, C26.49(C2×D4), C2.25(D4×D13), (C23×D13)⋊2C2, D26⋊C414C2, C222(C13⋊D4), (C2×C26).52C23, (C22×C26)⋊3C22, C23.D1310C2, (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

C1C2×C26 — C23⋊D26
C1C13C26C2×C26C22×D13C23×D13 — C23⋊D26
C13C2×C26 — C23⋊D26
C1C22C2×D4

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 [×2], C2 [×7], C4 [×3], C22, C22 [×2], C22 [×21], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], C13, C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, D13 [×4], C26, C26 [×2], C26 [×3], C22≀C2, Dic13 [×2], C52, D26 [×4], D26 [×12], C2×C26, C2×C26 [×2], C2×C26 [×5], C2×Dic13 [×2], C13⋊D4 [×4], C2×C52, D4×C13 [×2], C22×D13 [×2], C22×D13 [×6], C22×C26 [×2], D26⋊C4 [×2], C23.D13, C2×C13⋊D4 [×2], D4×C26, C23×D13, C23⋊D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], D13, C22≀C2, D26 [×3], C13⋊D4 [×2], C22×D13, D4×D13 [×2], C2×C13⋊D4, C23⋊D26

Smallest permutation representation of C23⋊D26
On 104 points
Generators in S104
(1 71)(2 59)(3 73)(4 61)(5 75)(6 63)(7 77)(8 65)(9 53)(10 67)(11 55)(12 69)(13 57)(14 60)(15 74)(16 62)(17 76)(18 64)(19 78)(20 66)(21 54)(22 68)(23 56)(24 70)(25 58)(26 72)(27 102)(28 90)(29 104)(30 92)(31 80)(32 94)(33 82)(34 96)(35 84)(36 98)(37 86)(38 100)(39 88)(40 85)(41 99)(42 87)(43 101)(44 89)(45 103)(46 91)(47 79)(48 93)(49 81)(50 95)(51 83)(52 97)
(1 37)(2 38)(3 39)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 40)(25 41)(26 42)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)
(1 25)(2 26)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 40)(37 41)(38 42)(39 43)(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 26)(13 25)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 52)(39 51)(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)| (1,71)(2,59)(3,73)(4,61)(5,75)(6,63)(7,77)(8,65)(9,53)(10,67)(11,55)(12,69)(13,57)(14,60)(15,74)(16,62)(17,76)(18,64)(19,78)(20,66)(21,54)(22,68)(23,56)(24,70)(25,58)(26,72)(27,102)(28,90)(29,104)(30,92)(31,80)(32,94)(33,82)(34,96)(35,84)(36,98)(37,86)(38,100)(39,88)(40,85)(41,99)(42,87)(43,101)(44,89)(45,103)(46,91)(47,79)(48,93)(49,81)(50,95)(51,83)(52,97), (1,37)(2,38)(3,39)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,40)(25,41)(26,42)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,40)(37,41)(38,42)(39,43)(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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,26)(13,25)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,52)(39,51)(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( (1,71)(2,59)(3,73)(4,61)(5,75)(6,63)(7,77)(8,65)(9,53)(10,67)(11,55)(12,69)(13,57)(14,60)(15,74)(16,62)(17,76)(18,64)(19,78)(20,66)(21,54)(22,68)(23,56)(24,70)(25,58)(26,72)(27,102)(28,90)(29,104)(30,92)(31,80)(32,94)(33,82)(34,96)(35,84)(36,98)(37,86)(38,100)(39,88)(40,85)(41,99)(42,87)(43,101)(44,89)(45,103)(46,91)(47,79)(48,93)(49,81)(50,95)(51,83)(52,97), (1,37)(2,38)(3,39)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,40)(25,41)(26,42)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,40)(37,41)(38,42)(39,43)(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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,26)(13,25)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,52)(39,51)(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([(1,71),(2,59),(3,73),(4,61),(5,75),(6,63),(7,77),(8,65),(9,53),(10,67),(11,55),(12,69),(13,57),(14,60),(15,74),(16,62),(17,76),(18,64),(19,78),(20,66),(21,54),(22,68),(23,56),(24,70),(25,58),(26,72),(27,102),(28,90),(29,104),(30,92),(31,80),(32,94),(33,82),(34,96),(35,84),(36,98),(37,86),(38,100),(39,88),(40,85),(41,99),(42,87),(43,101),(44,89),(45,103),(46,91),(47,79),(48,93),(49,81),(50,95),(51,83),(52,97)], [(1,37),(2,38),(3,39),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,40),(25,41),(26,42),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93)], [(1,25),(2,26),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,40),(37,41),(38,42),(39,43),(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,26),(13,25),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,52),(39,51),(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 2A2B2C2D2E2F2G2H2I2J4A4B4C13A···13F26A···26R26S···26AP52A···52L
order1222222222244413···1326···2626···2652···52
size111122426262626452522···22···24···44···4

74 irreducible representations

dim1111112222224
type++++++++++++
imageC1C2C2C2C2C2D4D4D13D26D26C13⋊D4D4×D13
kernelC23⋊D26D26⋊C4C23.D13C2×C13⋊D4D4×C26C23×D13D26C2×C26C2×D4C2×C4C23C22C2
# reps1212114266122412

Matrix representation of C23⋊D26 in GL4(𝔽53) generated by

481700
36500
005212
0001
,
52000
05200
0010
0001
,
1000
0100
00520
00052
,
515100
22800
0010
00952
,
121500
474100
00520
00441
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

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