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G = C22⋊D36order 288 = 25·32

The semidirect product of C22 and D36 acting via D36/C12=S3

non-abelian, soluble, monomial

Aliases: C22⋊D36, C12.2S4, C23.3D18, C3.(C4⋊S4), C4⋊(C3.S4), (C2×C6).D12, C3.A41D4, C6.18(C2×S4), (C22×C4)⋊2D9, (C22×C12).3S3, (C22×C6).15D6, (C2×C3.S4)⋊1C2, (C4×C3.A4)⋊1C2, C2.4(C2×C3.S4), (C2×C3.A4).3C22, SmallGroup(288,334)

Series: Derived Chief Lower central Upper central

C1C22C2×C3.A4 — C22⋊D36
C1C22C2×C6C3.A4C2×C3.A4C2×C3.S4 — C22⋊D36
C3.A4C2×C3.A4 — C22⋊D36
C1C2C4

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

Subgroups: 668 in 96 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22, C22 [×8], S3 [×2], C6, C6 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C9, Dic3 [×2], C12, C12, D6 [×6], C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D9 [×2], C18, D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, C36, C3.A4, D18 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, D36, C3.S4 [×2], C2×C3.A4, C127D4, C4×C3.A4, C2×C3.S4 [×2], C22⋊D36
Quotients: C1, C2 [×3], C22, S3, D4, D6, D9, D12, S4, D18, C2×S4, D36, C3.S4, C4⋊S4, C2×C3.S4, C22⋊D36

Character table of C22⋊D36

 class 12A2B2C2D2E34A4B4C4D6A6B6C9A9B9C12A12B12C12D18A18B18C36A36B36C36D36E36F
 size 1133363622636362668882266888888888
ρ1111111111111111111111111111111    trivial
ρ211111-11-1-11-1111111-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ31111-111-1-1-11111111-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111-1-11111111111111111111    linear of order 2
ρ522220022200222-1-1-12222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62222002-2-200222-1-1-1-2-2-2-2-1-1-1111111    orthogonal lifted from D6
ρ72-22-20020000-2-222220000-2-2-2000000    orthogonal lifted from D4
ρ8222200-1-2-200-1-1-1ζ989ζ9792ζ95941111ζ9792ζ9594ζ9899899594989979295949792    orthogonal lifted from D18
ρ9222200-1-2-200-1-1-1ζ9594ζ989ζ97921111ζ989ζ9792ζ95949594979295949899792989    orthogonal lifted from D18
ρ10222200-12200-1-1-1ζ9594ζ989ζ9792-1-1-1-1ζ989ζ9792ζ9594ζ9594ζ9792ζ9594ζ989ζ9792ζ989    orthogonal lifted from D9
ρ11222200-12200-1-1-1ζ9792ζ9594ζ989-1-1-1-1ζ9594ζ989ζ9792ζ9792ζ989ζ9792ζ9594ζ989ζ9594    orthogonal lifted from D9
ρ12222200-12200-1-1-1ζ989ζ9792ζ9594-1-1-1-1ζ9792ζ9594ζ989ζ989ζ9594ζ989ζ9792ζ9594ζ9792    orthogonal lifted from D9
ρ13222200-1-2-200-1-1-1ζ9792ζ9594ζ9891111ζ9594ζ989ζ97929792989979295949899594    orthogonal lifted from D18
ρ142-22-20020000-2-22-1-1-10000111-3-3333-3    orthogonal lifted from D12
ρ152-22-20020000-2-22-1-1-1000011133-3-3-33    orthogonal lifted from D12
ρ162-22-200-1000011-1ζ9792ζ9594ζ9893-33-395949899792ζ4ζ974ζ92ζ43ζ9843ζ94ζ974ζ92ζ4ζ954ζ9443ζ9843ζ94ζ954ζ94    orthogonal lifted from D36
ρ172-22-200-1000011-1ζ9792ζ9594ζ989-33-33959498997924ζ974ζ9243ζ9843ζ9ζ4ζ974ζ924ζ954ζ94ζ43ζ9843ζ9ζ4ζ954ζ94    orthogonal lifted from D36
ρ182-22-200-1000011-1ζ9594ζ989ζ97923-33-3989979295944ζ954ζ94ζ4ζ974ζ92ζ4ζ954ζ9443ζ9843ζ94ζ974ζ92ζ43ζ9843ζ9    orthogonal lifted from D36
ρ192-22-200-1000011-1ζ989ζ9792ζ95943-33-397929594989ζ43ζ9843ζ94ζ954ζ9443ζ9843ζ94ζ974ζ92ζ4ζ954ζ94ζ4ζ974ζ92    orthogonal lifted from D36
ρ202-22-200-1000011-1ζ989ζ9792ζ9594-33-339792959498943ζ9843ζ9ζ4ζ954ζ94ζ43ζ9843ζ9ζ4ζ974ζ924ζ954ζ944ζ974ζ92    orthogonal lifted from D36
ρ212-22-200-1000011-1ζ9594ζ989ζ9792-33-3398997929594ζ4ζ954ζ944ζ974ζ924ζ954ζ94ζ43ζ9843ζ9ζ4ζ974ζ9243ζ9843ζ9    orthogonal lifted from D36
ρ2233-1-1-113-311-13-1-1000-3-311000000000    orthogonal lifted from C2×S4
ρ2333-1-11133-1-1-13-1-100033-1-1000000000    orthogonal lifted from S4
ρ2433-1-1-1-133-1113-1-100033-1-1000000000    orthogonal lifted from S4
ρ2533-1-11-13-31-113-1-1000-3-311000000000    orthogonal lifted from C2×S4
ρ266-6-220060000-62-20000000000000000    orthogonal lifted from C4⋊S4
ρ2766-2-200-36-200-311000-3-311000000000    orthogonal lifted from C3.S4
ρ2866-2-200-3-6200-31100033-1-1000000000    orthogonal lifted from C2×C3.S4
ρ296-6-2200-300003-11000-33333-3000000000    orthogonal faithful
ρ306-6-2200-300003-1100033-33-33000000000    orthogonal faithful

Smallest permutation representation of C22⋊D36
On 36 points
Generators in S36
(1 19)(2 20)(4 22)(5 23)(7 25)(8 26)(10 28)(11 29)(13 31)(14 32)(16 34)(17 35)
(1 19)(3 21)(4 22)(6 24)(7 25)(9 27)(10 28)(12 30)(13 31)(15 33)(16 34)(18 36)
(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)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)

G:=sub<Sym(36)| (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,28)(12,30)(13,31)(15,33)(16,34)(18,36), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)>;

G:=Group( (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,28)(12,30)(13,31)(15,33)(16,34)(18,36), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33) );

G=PermutationGroup([(1,19),(2,20),(4,22),(5,23),(7,25),(8,26),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35)], [(1,19),(3,21),(4,22),(6,24),(7,25),(9,27),(10,28),(12,30),(13,31),(15,33),(16,34),(18,36)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(28,36),(29,35),(30,34),(31,33)])

Matrix representation of C22⋊D36 in GL5(𝔽37)

10000
01000
000361
000360
001360
,
10000
01000
003600
003601
003610
,
277000
3027000
003610
003600
003601
,
1429000
2923000
003600
003610
003601

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[27,30,0,0,0,7,27,0,0,0,0,0,36,36,36,0,0,1,0,0,0,0,0,0,1],[14,29,0,0,0,29,23,0,0,0,0,0,36,36,36,0,0,0,1,0,0,0,0,0,1] >;

C22⋊D36 in GAP, Magma, Sage, TeX

C_2^2\rtimes D_{36}
% in TeX

G:=Group("C2^2:D36");
// GroupNames label

G:=SmallGroup(288,334);
// by ID

G=gap.SmallGroup(288,334);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,1123,192,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,d*a*d=c*b*c^-1=a*b=b*a,c*a*c^-1=b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of C22⋊D36 in TeX

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