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G = C223D36order 288 = 25·32

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D184D4, C223D36, C23.20D18, C91C22≀C2, (C2×C4)⋊1D18, (C2×C18)⋊1D4, C2.7(D4×D9), D18⋊C44C2, (C2×D36)⋊2C2, C22⋊C42D9, 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

C1C2×C18 — C223D36
C1C3C9C18C2×C18C22×D9C23×D9 — C223D36
C9C2×C18 — C223D36
C1C22C22⋊C4

Generators and relations for C223D36
 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, C223D36
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], C223D36

Smallest permutation representation of C223D36
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
kernelC223D36D18⋊C4C9×C22⋊C4C2×D36C2×C9⋊D4C23×D9C3×C22⋊C4D18C2×C18C2×C12C22×C6C22⋊C4C2×C6C2×C4C23C22C6C2
# reps1212111422134631226

Matrix representation of C223D36 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] >;

C223D36 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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