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G = C22⋊3D36  order 288 = 25·32

The semidirect product of C22 and D36 acting via D36/D18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D18⋊4D4, C22⋊3D36, C23.20D18, C9⋊1C22≀C2, (C2×C4)⋊1D18, (C2×C18)⋊1D4, C2.7(D4×D9), D18⋊C4⋊4C2, (C2×D36)⋊2C2, C22⋊C4⋊2D9, C18.5(C2×D4), C6.79(S3×D4), (C2×C12).2D6, C2.7(C2×D36), (C2×C6).4D12, (C2×C36)⋊1C22, C3.(D6⋊D4), C6.34(C2×D12), (C23×D9)⋊1C2, (C22×C6).42D6, (C2×C18).23C23, (C2×Dic9)⋊1C22, (C22×D9)⋊1C22, C22.41(C22×D9), (C22×C18).12C22, (C2×C9⋊D4)⋊1C2, (C9×C22⋊C4)⋊3C2, (C3×C22⋊C4).3S3, (C2×C6).180(C22×S3), SmallGroup(288,92)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C18 — C22⋊3D36
Chief series
C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C23×D9 — C22⋊3D36
Lower central
C9 — C2×C18 — C22⋊3D36
Upper central
C1 — C22 — C22⋊C4

Generators and relations for C22⋊3D36
 G = < a,b,c,d | a2=b2=c36=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1148 in 195 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×5], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], C9, Dic3, C12 [×2], D6 [×19], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, D9 [×5], C18, C18 [×2], C18 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3 [×9], C22×C6, C22≀C2, Dic9, C36 [×2], D18 [×4], D18 [×15], C2×C18, C2×C18 [×2], C2×C18 [×2], D6⋊C4 [×2], C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, D36 [×4], C2×Dic9, C9⋊D4 [×2], C2×C36 [×2], C22×D9, C22×D9 [×2], C22×D9 [×6], C22×C18, D6⋊D4, D18⋊C4 [×2], C9×C22⋊C4, C2×D36 [×2], C2×C9⋊D4, C23×D9, C22⋊3D36
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D9, D12 [×2], C22×S3, C22≀C2, D18 [×3], C2×D12, S3×D4 [×2], D36 [×2], C22×D9, D6⋊D4, C2×D36, D4×D9 [×2], C22⋊3D36

Smallest permutation representation of C22⋊3D36
►On 72 points
Generators in S72
(1 43)(3 45)(5 47)(7 49)(9 51)(11 53)(13 55)(15 57)(17 59)(19 61)(21 63)(23 65)(25 67)(27 69)(29 71)(31 37)(33 39)(35 41)

(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)

(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)

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122222222223444666669991212121218···1818···1836···36
size1111221818181836244362224422244442···24···44···4

54 irreducible representations

dim111111222222222244
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D9D12D18D18D36S3×D4D4×D9
kernelC22⋊3D36D18⋊C4C9×C22⋊C4C2×D36C2×C9⋊D4C23×D9C3×C22⋊C4D18C2×C18C2×C12C22×C6C22⋊C4C2×C6C2×C4C23C22C6C2
# reps1212111422134631226

Matrix representation of C22⋊3D36 ►in GL4(𝔽37) generated by

36000
0100
0010
0001
,
36000
03600
0010
0001
,
0100
36000
0048
002912
,
03600
36000
003210
0055
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,1,0,0,0,0,0,4,29,0,0,8,12],[0,36,0,0,36,0,0,0,0,0,32,5,0,0,10,5] >;
 

C22⋊3D36 in GAP, Magma, Sage, TeX 

C_2^2\rtimes_3D_{36}
 
% in TeX
 
G:=Group("C2^2:3D36");
 
// GroupNames label
 
G:=SmallGroup(288,92);
 
// by ID
 
G=gap.SmallGroup(288,92);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,6725,292,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^36=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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