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## G = C23×C13⋊C4order 416 = 25·13

### Direct product of C23 and C13⋊C4

Aliases: C23×C13⋊C4, D13.C24, D26.15C23, C13⋊(C23×C4), C26⋊(C22×C4), D269(C2×C4), D13⋊(C22×C4), (C22×C26)⋊5C4, (C22×D13)⋊6C4, (C23×D13).4C2, (C22×D13).40C22, (C2×C26)⋊3(C2×C4), SmallGroup(416,233)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C23×C13⋊C4
 Chief series C1 — C13 — D13 — C13⋊C4 — C2×C13⋊C4 — C22×C13⋊C4 — C23×C13⋊C4
 Lower central C13 — C23×C13⋊C4
 Upper central C1 — C23

Generators and relations for C23×C13⋊C4
G = < a,b,c,d,e | a2=b2=c2=d13=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 1460 in 236 conjugacy classes, 134 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C23, C23, C13, C22×C4, C24, D13, D13, C26, C23×C4, C13⋊C4, D26, C2×C26, C2×C13⋊C4, C22×D13, C22×C26, C22×C13⋊C4, C23×D13, C23×C13⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, C13⋊C4, C2×C13⋊C4, C22×C13⋊C4, C23×C13⋊C4

Smallest permutation representation of C23×C13⋊C4
On 104 points
Generators in S104
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(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 66)(2 74 13 71)(3 69 12 76)(4 77 11 68)(5 72 10 73)(6 67 9 78)(7 75 8 70)(14 53)(15 61 26 58)(16 56 25 63)(17 64 24 55)(18 59 23 60)(19 54 22 65)(20 62 21 57)(27 92)(28 100 39 97)(29 95 38 102)(30 103 37 94)(31 98 36 99)(32 93 35 104)(33 101 34 96)(40 79)(41 87 52 84)(42 82 51 89)(43 90 50 81)(44 85 49 86)(45 80 48 91)(46 88 47 83)

G:=sub<Sym(104)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(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), (1,66)(2,74,13,71)(3,69,12,76)(4,77,11,68)(5,72,10,73)(6,67,9,78)(7,75,8,70)(14,53)(15,61,26,58)(16,56,25,63)(17,64,24,55)(18,59,23,60)(19,54,22,65)(20,62,21,57)(27,92)(28,100,39,97)(29,95,38,102)(30,103,37,94)(31,98,36,99)(32,93,35,104)(33,101,34,96)(40,79)(41,87,52,84)(42,82,51,89)(43,90,50,81)(44,85,49,86)(45,80,48,91)(46,88,47,83)>;

G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(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), (1,66)(2,74,13,71)(3,69,12,76)(4,77,11,68)(5,72,10,73)(6,67,9,78)(7,75,8,70)(14,53)(15,61,26,58)(16,56,25,63)(17,64,24,55)(18,59,23,60)(19,54,22,65)(20,62,21,57)(27,92)(28,100,39,97)(29,95,38,102)(30,103,37,94)(31,98,36,99)(32,93,35,104)(33,101,34,96)(40,79)(41,87,52,84)(42,82,51,89)(43,90,50,81)(44,85,49,86)(45,80,48,91)(46,88,47,83) );

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4P 13A 13B 13C 26A ··· 26U order 1 2 ··· 2 2 ··· 2 4 ··· 4 13 13 13 26 ··· 26 size 1 1 ··· 1 13 ··· 13 13 ··· 13 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 4 4 type + + + + + image C1 C2 C2 C4 C4 C13⋊C4 C2×C13⋊C4 kernel C23×C13⋊C4 C22×C13⋊C4 C23×D13 C22×D13 C22×C26 C23 C22 # reps 1 14 1 14 2 3 21

Matrix representation of C23×C13⋊C4 in GL7(𝔽53)

 1 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 52 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 52 0 0 0 0 0 0 0 52 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 48 19 33 52 0 0 0 31 45 14 6 0 0 0 43 39 13 51 0 0 0 36 1 39 39
,
 23 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 22 35 25 12 0 0 0 32 16 35 17 0 0 0 41 10 43 39 0 0 0 0 36 49 25

G:=sub<GL(7,GF(53))| [1,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[52,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[52,0,0,0,0,0,0,0,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,48,31,43,36,0,0,0,19,45,39,1,0,0,0,33,14,13,39,0,0,0,52,6,51,39],[23,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,22,32,41,0,0,0,0,35,16,10,36,0,0,0,25,35,43,49,0,0,0,12,17,39,25] >;

C23×C13⋊C4 in GAP, Magma, Sage, TeX

C_2^3\times C_{13}\rtimes C_4
% in TeX

G:=Group("C2^3xC13:C4");
// GroupNames label

G:=SmallGroup(416,233);
// by ID

G=gap.SmallGroup(416,233);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,9221,893]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^13=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*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^5>;
// generators/relations

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