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G = C36⋊D6order 432 = 24·33

2nd semidirect product of C36 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C362D6, D182D6, D62D18, D364S3, D124D9, C122D18, C9⋊S31D4, C92(S3×D4), C32(D4×D9), C43(S3×D9), C12.20S32, (C3×D36)⋊8C2, (C9×D12)⋊6C2, (S3×C6).4D6, D6⋊D93C2, (C3×C36)⋊3C22, (C3×D12).8S3, (C6×D9)⋊2C22, C32.3(S3×D4), (S3×C18)⋊2C22, (C3×C12).100D6, C9⋊Dic34C22, C6.14(C22×D9), C3.1(D6⋊D6), C18.14(C22×S3), (C3×C18).14C23, (C2×S3×D9)⋊3C2, (C3×C9)⋊3(C2×D4), (C4×C9⋊S3)⋊3C2, C6.33(C2×S32), C2.17(C2×S3×D9), (C2×C9⋊S3).10C22, (C3×C6).82(C22×S3), SmallGroup(432,293)

Series: Derived Chief Lower central Upper central

C1C3×C18 — C36⋊D6
C1C3C32C3×C9C3×C18S3×C18C2×S3×D9 — C36⋊D6
C3×C9C3×C18 — C36⋊D6
C1C2C4

Generators and relations for C36⋊D6
 G = < a,b,c | a36=b6=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >

Subgroups: 1352 in 178 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×10], C6 [×2], C6 [×5], C2×C4, D4 [×4], C23 [×2], C9, C9, C32, Dic3 [×3], C12 [×2], C12, D6 [×2], D6 [×13], C2×C6 [×4], C2×D4, D9 [×6], C18, C18 [×3], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×3], D12, D12, C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C3×C9, Dic9 [×2], C36, C36, D18 [×2], D18 [×6], C2×C18 [×2], C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3, S3×D4 [×2], C3×D9 [×2], S3×C9 [×2], C9⋊S3 [×2], C3×C18, C4×D9 [×2], D36, C9⋊D4 [×2], D4×C9, C22×D9 [×2], D6⋊S3 [×2], C3×D12, C3×D12, C4×C3⋊S3, C2×S32 [×2], C9⋊Dic3, C3×C36, S3×D9 [×4], C6×D9 [×2], S3×C18 [×2], C2×C9⋊S3, D4×D9, D6⋊D6, D6⋊D9 [×2], C3×D36, C9×D12, C4×C9⋊S3, C2×S3×D9 [×2], C36⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D9, C22×S3 [×2], D18 [×3], S32, S3×D4 [×2], C22×D9, C2×S32, S3×D9, D4×D9, D6⋊D6, C2×S3×D9, C36⋊D6

Smallest permutation representation of C36⋊D6
On 72 points
Generators in S72
(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 49 13 37 25 61)(2 48 14 72 26 60)(3 47 15 71 27 59)(4 46 16 70 28 58)(5 45 17 69 29 57)(6 44 18 68 30 56)(7 43 19 67 31 55)(8 42 20 66 32 54)(9 41 21 65 33 53)(10 40 22 64 34 52)(11 39 23 63 35 51)(12 38 24 62 36 50)
(1 13)(2 30)(3 11)(4 28)(5 9)(6 26)(8 24)(10 22)(12 20)(14 18)(15 35)(17 33)(19 31)(21 29)(23 27)(32 36)(37 61)(38 42)(39 59)(41 57)(43 55)(44 72)(45 53)(46 70)(47 51)(48 68)(50 66)(52 64)(54 62)(56 60)(63 71)(65 69)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G9A9B9C9D9E9F12A12B12C12D18A18B18C18D18E18F18G···18L36A···36I
order122222223334466666669999991212121218181818181818···1836···36
size11661818272722425422412123636222444444422244412···124···4

51 irreducible representations

dim1111112222222222444444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6D6D9D18D18S32S3×D4S3×D4C2×S32S3×D9D4×D9D6⋊D6C2×S3×D9C36⋊D6
kernelC36⋊D6D6⋊D9C3×D36C9×D12C4×C9⋊S3C2×S3×D9D36C3×D12C9⋊S3C36D18C3×C12S3×C6D12C12D6C12C9C32C6C4C3C3C2C1
# reps1211121121212336111133236

Matrix representation of C36⋊D6 in GL6(𝔽37)

130000
24360000
001000
000100
00002011
00002631
,
15320000
30220000
0003600
001100
000001
000010
,
3600000
0360000
000100
001000
000001
000010

G:=sub<GL(6,GF(37))| [1,24,0,0,0,0,3,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,26,0,0,0,0,11,31],[15,30,0,0,0,0,32,22,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C36⋊D6 in GAP, Magma, Sage, TeX

C_{36}\rtimes D_6
% in TeX

G:=Group("C36:D6");
// GroupNames label

G:=SmallGroup(432,293);
// by ID

G=gap.SmallGroup(432,293);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c|a^36=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations

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