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G = D48D22order 352 = 25·11

4th semidirect product of D4 and D22 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D48D22, Q87D22, D4411C22, C22.12C24, C44.26C23, D22.7C23, C1122+ 1+4, Dic2212C22, Dic11.7C23, (C2×C4)⋊4D22, C4○D43D11, (D4×D11)⋊5C2, (C2×D44)⋊13C2, (C2×C44)⋊5C22, D44⋊C25C2, D445C28C2, (D4×C11)⋊9C22, (C4×D11)⋊2C22, C11⋊D45C22, (C2×C22).4C23, (Q8×C11)⋊8C22, C4.33(C22×D11), C2.13(C23×D11), (C22×D11)⋊4C22, C22.3(C22×D11), (C11×C4○D4)⋊4C2, SmallGroup(352,184)

Series: Derived Chief Lower central Upper central

C1C22 — D48D22
C1C11C22D22C22×D11D4×D11 — D48D22
C11C22 — D48D22
C1C2C4○D4

Generators and relations for D48D22
 G = < a,b,c,d | a4=b2=c22=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >

Subgroups: 1090 in 166 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C11, C2×D4, C4○D4, C4○D4, D11, C22, C22, 2+ 1+4, Dic11, C44, C44, D22, D22, C2×C22, Dic22, C4×D11, D44, C11⋊D4, C2×C44, D4×C11, Q8×C11, C22×D11, C2×D44, D445C2, D4×D11, D44⋊C2, C11×C4○D4, D48D22
Quotients: C1, C2, C22, C23, C24, D11, 2+ 1+4, D22, C22×D11, C23×D11, D48D22

Smallest permutation representation of D48D22
On 88 points
Generators in S88
(1 32 69 46)(2 33 70 47)(3 34 71 48)(4 35 72 49)(5 36 73 50)(6 37 74 51)(7 38 75 52)(8 39 76 53)(9 40 77 54)(10 41 78 55)(11 42 79 56)(12 43 80 57)(13 44 81 58)(14 23 82 59)(15 24 83 60)(16 25 84 61)(17 26 85 62)(18 27 86 63)(19 28 87 64)(20 29 88 65)(21 30 67 66)(22 31 68 45)
(1 46)(2 33)(3 48)(4 35)(5 50)(6 37)(7 52)(8 39)(9 54)(10 41)(11 56)(12 43)(13 58)(14 23)(15 60)(16 25)(17 62)(18 27)(19 64)(20 29)(21 66)(22 31)(24 83)(26 85)(28 87)(30 67)(32 69)(34 71)(36 73)(38 75)(40 77)(42 79)(44 81)(45 68)(47 70)(49 72)(51 74)(53 76)(55 78)(57 80)(59 82)(61 84)(63 86)(65 88)
(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)
(1 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)

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

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

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

67 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F11A···11E22A···22E22F···22T44A···44J44K···44Y
order122222···244444411···1122···2222···2244···4444···44
size1122222···22222222222···22···24···42···24···4

67 irreducible representations

dim111111222244
type++++++++++++
imageC1C2C2C2C2C2D11D22D22D222+ 1+4D48D22
kernelD48D22C2×D44D445C2D4×D11D44⋊C2C11×C4○D4C4○D4C2×C4D4Q8C11C1
# reps133621515155110

Matrix representation of D48D22 in GL4(𝔽89) generated by

85147848
3845648
1403175
27144517
,
85147848
3845648
71143175
16584517
,
49515083
23537481
5108438
51517681
,
6688250
16337357
6002829
34685522
G:=sub<GL(4,GF(89))| [85,38,14,27,14,45,0,14,78,64,31,45,48,8,75,17],[85,38,71,16,14,45,14,58,78,64,31,45,48,8,75,17],[49,23,51,51,51,53,0,51,50,74,84,76,83,81,38,81],[6,16,60,34,68,33,0,68,82,73,28,55,50,57,29,22] >;

D48D22 in GAP, Magma, Sage, TeX

D_4\rtimes_8D_{22}
% in TeX

G:=Group("D4:8D22");
// GroupNames label

G:=SmallGroup(352,184);
// by ID

G=gap.SmallGroup(352,184);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,188,579,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^22=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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