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G = C22⋊C4×D13order 416 = 25·13

Direct product of C22⋊C4 and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22⋊C4×D13, D26.21D4, C23.13D26, (C2×C4)⋊5D26, D266(C2×C4), C2.1(D4×D13), (C2×C52)⋊6C22, C26.17(C2×D4), C223(C4×D13), D26⋊C49C2, (C22×D13)⋊3C4, C23.D133C2, (C2×C26).21C23, C26.19(C22×C4), (C23×D13).1C2, (C2×Dic13)⋊5C22, (C22×C26).10C22, C22.13(C22×D13), (C22×D13).42C22, (C2×C4×D13)⋊8C2, C2.8(C2×C4×D13), (C2×C26)⋊4(C2×C4), C132(C2×C22⋊C4), (C13×C22⋊C4)⋊8C2, SmallGroup(416,101)

Series: Derived Chief Lower central Upper central

C1C26 — C22⋊C4×D13
C1C13C26C2×C26C22×D13C23×D13 — C22⋊C4×D13
C13C26 — C22⋊C4×D13
C1C22C22⋊C4

Generators and relations for C22⋊C4×D13
 G = < a,b,c,d,e | a2=b2=c4=d13=e2=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1040 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C22, C22 [×2], C22 [×20], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C13, C22⋊C4, C22⋊C4 [×3], C22×C4 [×2], C24, D13 [×4], D13 [×2], C26, C26 [×2], C26 [×2], C2×C22⋊C4, Dic13 [×2], C52 [×2], D26 [×8], D26 [×10], C2×C26, C2×C26 [×2], C2×C26 [×2], C4×D13 [×4], C2×Dic13 [×2], C2×C52 [×2], C22×D13 [×2], C22×D13 [×4], C22×D13 [×4], C22×C26, D26⋊C4 [×2], C23.D13, C13×C22⋊C4, C2×C4×D13 [×2], C23×D13, C22⋊C4×D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D13, C2×C22⋊C4, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D4×D13 [×2], C22⋊C4×D13

Smallest permutation representation of C22⋊C4×D13
On 104 points
Generators in S104
(1 39)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(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 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(65 92)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(78 85)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(27 51)(28 52)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)(79 103)(80 104)(81 92)(82 93)(83 94)(84 95)(85 96)(86 97)(87 98)(88 99)(89 100)(90 101)(91 102)
(1 83 39 54)(2 84 27 55)(3 85 28 56)(4 86 29 57)(5 87 30 58)(6 88 31 59)(7 89 32 60)(8 90 33 61)(9 91 34 62)(10 79 35 63)(11 80 36 64)(12 81 37 65)(13 82 38 53)(14 99 42 68)(15 100 43 69)(16 101 44 70)(17 102 45 71)(18 103 46 72)(19 104 47 73)(20 92 48 74)(21 93 49 75)(22 94 50 76)(23 95 51 77)(24 96 52 78)(25 97 40 66)(26 98 41 67)
(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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 52)(12 51)(13 50)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(53 94)(54 93)(55 92)(56 104)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 79)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H13A···13F26A···26R26S···26AD52A···52X
order1222222222224444444413···1326···2626···2652···52
size1111221313131326262222262626262···22···24···44···4

80 irreducible representations

dim1111111222224
type+++++++++++
imageC1C2C2C2C2C2C4D4D13D26D26C4×D13D4×D13
kernelC22⋊C4×D13D26⋊C4C23.D13C13×C22⋊C4C2×C4×D13C23×D13C22×D13D26C22⋊C4C2×C4C23C22C2
# reps1211218461262412

Matrix representation of C22⋊C4×D13 in GL4(𝔽53) generated by

52000
05200
0010
004952
,
1000
0100
00520
00052
,
30000
03000
004951
00344
,
355100
524400
0010
0001
,
184800
13500
00520
00052
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,1,49,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[30,0,0,0,0,30,0,0,0,0,49,34,0,0,51,4],[35,52,0,0,51,44,0,0,0,0,1,0,0,0,0,1],[18,1,0,0,48,35,0,0,0,0,52,0,0,0,0,52] >;

C22⋊C4×D13 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times D_{13}
% in TeX

G:=Group("C2^2:C4xD13");
// GroupNames label

G:=SmallGroup(416,101);
// by ID

G=gap.SmallGroup(416,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,50,13829]);
// Polycyclic

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

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