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G = C22⋊D45  order 360 = 23·32·5

The semidirect product of C22 and D45 acting via D45/C15=S3

non-abelian, soluble, monomial

Aliases: C22⋊D45, C15.1S4, C3.(C5⋊S4), C5⋊(C3.S4), C3.A4⋊D5, (C2×C6).D15, (C2×C10)⋊2D9, (C2×C30).1S3, (C5×C3.A4)⋊1C2, SmallGroup(360,41)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C3.A4 — C22⋊D45
Chief series
C1 — C22 — C2×C6 — C2×C30 — C5×C3.A4 — C22⋊D45
Lower central
C5×C3.A4 — C22⋊D45
Upper central
C1

Generators and relations for C22⋊D45
 G = < a,b,c,d | a2=b2=c45=d2=1, cac-1=ab=ba, dad=b, cbc-1=dbd=a, dcd=c-1 >

C1
3C2
90C2
C3
C5
C22
45C4
45C22
3C6
30S3
4C9
3C10
18D5
C15
45D4
C2×C6
15D6
15Dic3
20D9
C2×C10
9Dic5
9D10
3C30
6D15
4C45
15C3⋊D4
C3.A4
9C5⋊D4
C2×C30
3D30
3Dic15
4D45
5C3.S4
3C15⋊7D4
C5×C3.A4
C22⋊D45

Smallest permutation representation of C22⋊D45
►On 90 points
Generators in S90
(1 84)(3 86)(4 87)(6 89)(7 90)(9 47)(10 48)(12 50)(13 51)(15 53)(16 54)(18 56)(19 57)(21 59)(22 60)(24 62)(25 63)(27 65)(28 66)(30 68)(31 69)(33 71)(34 72)(36 74)(37 75)(39 77)(40 78)(42 80)(43 81)(45 83)

(1 84)(2 85)(4 87)(5 88)(7 90)(8 46)(10 48)(11 49)(13 51)(14 52)(16 54)(17 55)(19 57)(20 58)(22 60)(23 61)(25 63)(26 64)(28 66)(29 67)(31 69)(32 70)(34 72)(35 73)(37 75)(38 76)(40 78)(41 79)(43 81)(44 82)

(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)

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

33 conjugacy classes

class 1 2A2B 3  4 5A5B 6 9A9B9C10A10B15A15B15C15D30A30B30C30D45A···45L
order122345569991010151515153030303045···45
size139029022688866222266668···8

33 irreducible representations

dim11222223666
type+++++++++++
imageC1C2S3D5D9D15D45S4C3.S4C5⋊S4C22⋊D45
kernelC22⋊D45C5×C3.A4C2×C30C3.A4C2×C10C2×C6C22C15C5C3C1
# reps111234122124

Matrix representation of C22⋊D45 ►in GL7(𝔽181)

1000000
0100000
0010000
0001000
000018000
000001800
0000001
,
1000000
0100000
0010000
0001000
000018000
0000010
000000180
,
17713100000
5012700000
0016684000
002328000
0000010
0000001
0000100
,
446200000
10613700000
0071148000
0054110000
0000100
0000001
0000010

G:=sub<GL(7,GF(181))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,180,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,180],[177,50,0,0,0,0,0,131,127,0,0,0,0,0,0,0,166,23,0,0,0,0,0,84,28,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[44,106,0,0,0,0,0,62,137,0,0,0,0,0,0,0,71,54,0,0,0,0,0,148,110,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
 

C22⋊D45 in GAP, Magma, Sage, TeX 

C_2^2\rtimes D_{45}
 
% in TeX
 
G:=Group("C2^2:D45");
 
// GroupNames label
 
G:=SmallGroup(360,41);
 
// by ID
 
G=gap.SmallGroup(360,41);
 
# by ID
 
G:=PCGroup([6,-2,-3,-5,-3,-2,2,409,367,434,1443,5404,2710,3245,4871]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^45=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=b,c*b*c^-1=d*b*d=a,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C22⋊D45 in TeX

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