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G = C22⋊D52order 416 = 25·13

The semidirect product of C22 and D52 acting via D52/D26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D264D4, C222D52, C23.15D26, (C2×C4)⋊1D26, (C2×C26)⋊1D4, C131C22≀C2, (C2×D52)⋊2C2, C2.7(C2×D52), C26.5(C2×D4), C2.7(D4×D13), (C2×C52)⋊1C22, C22⋊C42D13, D26⋊C44C2, (C23×D13)⋊1C2, (C2×C26).23C23, (C2×Dic13)⋊1C22, (C22×D13)⋊1C22, (C22×C26).12C22, C22.41(C22×D13), (C2×C13⋊D4)⋊1C2, (C13×C22⋊C4)⋊3C2, SmallGroup(416,103)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C22⋊D52
C1C13C26C2×C26C22×D13C23×D13 — C22⋊D52
C13C2×C26 — C22⋊D52
C1C22C22⋊C4

Generators and relations for C22⋊D52
 G = < a,b,c,d | a2=b2=c52=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1208 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×3], C22, C22 [×2], C22 [×21], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], C13, C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, D13 [×5], C26, C26 [×2], C26 [×2], C22≀C2, Dic13, C52 [×2], D26 [×4], D26 [×15], C2×C26, C2×C26 [×2], C2×C26 [×2], D52 [×4], C2×Dic13, C13⋊D4 [×2], C2×C52 [×2], C22×D13, C22×D13 [×2], C22×D13 [×6], C22×C26, D26⋊C4 [×2], C13×C22⋊C4, C2×D52 [×2], C2×C13⋊D4, C23×D13, C22⋊D52
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], D13, C22≀C2, D26 [×3], D52 [×2], C22×D13, C2×D52, D4×D13 [×2], C22⋊D52

Smallest permutation representation of C22⋊D52
On 104 points
Generators in S104
(1 27)(2 103)(3 29)(4 53)(5 31)(6 55)(7 33)(8 57)(9 35)(10 59)(11 37)(12 61)(13 39)(14 63)(15 41)(16 65)(17 43)(18 67)(19 45)(20 69)(21 47)(22 71)(23 49)(24 73)(25 51)(26 75)(28 77)(30 79)(32 81)(34 83)(36 85)(38 87)(40 89)(42 91)(44 93)(46 95)(48 97)(50 99)(52 101)(54 80)(56 82)(58 84)(60 86)(62 88)(64 90)(66 92)(68 94)(70 96)(72 98)(74 100)(76 102)(78 104)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)
(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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 104)(25 103)(26 102)(27 101)(28 100)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C13A···13F26A···26R26S···26AD52A···52X
order1222222222244413···1326···2626···2652···52
size111122262626265244522···22···24···44···4

74 irreducible representations

dim1111112222224
type+++++++++++++
imageC1C2C2C2C2C2D4D4D13D26D26D52D4×D13
kernelC22⋊D52D26⋊C4C13×C22⋊C4C2×D52C2×C13⋊D4C23×D13D26C2×C26C22⋊C4C2×C4C23C22C2
# reps1212114261262412

Matrix representation of C22⋊D52 in GL4(𝔽53) generated by

52000
05200
00520
00291
,
1000
0100
00520
00052
,
281800
454600
002138
003332
,
32300
255000
003215
00621
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,29,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[28,45,0,0,18,46,0,0,0,0,21,33,0,0,38,32],[3,25,0,0,23,50,0,0,0,0,32,6,0,0,15,21] >;

C22⋊D52 in GAP, Magma, Sage, TeX

C_2^2\rtimes D_{52}
% in TeX

G:=Group("C2^2:D52");
// GroupNames label

G:=SmallGroup(416,103);
// by ID

G=gap.SmallGroup(416,103);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,50,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^52=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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