Copied to
clipboard

G = C2×C9⋊D12order 432 = 24·33

Direct product of C2 and C9⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×C9⋊D12, D66D18, C182D12, Dic94D6, C62.69D6, C93(C2×D12), (C3×C18)⋊3D4, C61(C9⋊D4), (C6×Dic9)⋊8C2, (C2×Dic9)⋊4S3, (S3×C6).33D6, (C2×C6).22D18, (C2×C18).22D6, (C22×S3)⋊3D9, (S3×C18)⋊9C22, C22.17(S3×D9), C6.24(C22×D9), (C3×C18).24C23, (C6×C18).18C22, C18.24(C22×S3), (C3×Dic9)⋊7C22, C6.21(C3⋊D12), (C3×C9)⋊6(C2×D4), (S3×C2×C18)⋊5C2, C6.43(C2×S32), (C2×C6).28S32, C31(C2×C9⋊D4), (S3×C2×C6).5S3, C2.24(C2×S3×D9), (C22×C9⋊S3)⋊2C2, (C2×C9⋊S3)⋊6C22, C3.2(C2×C3⋊D12), C32.3(C2×C3⋊D4), (C3×C6).55(C3⋊D4), (C3×C6).92(C22×S3), SmallGroup(432,312)

Series: Derived Chief Lower central Upper central

C1C3×C18 — C2×C9⋊D12
C1C3C32C3×C9C3×C18S3×C18C9⋊D12 — C2×C9⋊D12
C3×C9C3×C18 — C2×C9⋊D12
C1C22

Generators and relations for C2×C9⋊D12
 G = < a,b,c,d | a2=b9=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1356 in 194 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C3×C9, Dic9, D18, C2×C18, C2×C18, C3×Dic3, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×D12, C2×C3⋊D4, S3×C9, C9⋊S3, C3×C18, C3×C18, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, C3×Dic9, S3×C18, S3×C18, C2×C9⋊S3, C2×C9⋊S3, C6×C18, C2×C9⋊D4, C2×C3⋊D12, C9⋊D12, C6×Dic9, S3×C2×C18, C22×C9⋊S3, C2×C9⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C3⋊D4, C22×S3, D18, S32, C2×D12, C2×C3⋊D4, C9⋊D4, C22×D9, C3⋊D12, C2×S32, S3×D9, C2×C9⋊D4, C2×C3⋊D12, C9⋊D12, C2×S3×D9, C2×C9⋊D12

Smallest permutation representation of C2×C9⋊D12
On 72 points
Generators in S72
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 72)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)
(1 62 13 9 70 21 5 66 17)(2 18 67 6 22 71 10 14 63)(3 64 15 11 72 23 7 68 19)(4 20 69 8 24 61 12 16 65)(25 46 59 29 38 51 33 42 55)(26 56 43 34 52 39 30 60 47)(27 48 49 31 40 53 35 44 57)(28 58 45 36 54 41 32 50 37)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 72)(22 71)(23 70)(24 69)(25 38)(26 37)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(49 57)(50 56)(51 55)(52 54)(58 60)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A···6F6G6H6I6J6K6L6M9A9B9C9D9E9F12A12B12C12D18A···18I18J···18R18S···18AD
order12222222333446···666666669999991212121218···1818···1818···18
size111166545422418182···24446666222444181818182···24···46···6

66 irreducible representations

dim111112222222222222444444
type++++++++++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6D6D6D9D12C3⋊D4D18D18C9⋊D4S32C3⋊D12C2×S32S3×D9C9⋊D12C2×S3×D9
kernelC2×C9⋊D12C9⋊D12C6×Dic9S3×C2×C18C22×C9⋊S3C2×Dic9S3×C2×C6C3×C18Dic9C2×C18S3×C6C62C22×S3C18C3×C6D6C2×C6C6C2×C6C6C6C22C2C2
# reps1411111221213446312121363

Matrix representation of C2×C9⋊D12 in GL6(𝔽37)

100000
010000
0036000
0003600
000010
000001
,
100000
010000
001000
000100
0000626
00001117
,
27320000
5320000
0010500
0032500
000010
00003636
,
0360000
3600000
0003600
0036000
0000360
000011

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,11,0,0,0,0,26,17],[27,5,0,0,0,0,32,32,0,0,0,0,0,0,10,32,0,0,0,0,5,5,0,0,0,0,0,0,1,36,0,0,0,0,0,36],[0,36,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,1] >;

C2×C9⋊D12 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes D_{12}
% in TeX

G:=Group("C2xC9:D12");
// GroupNames label

G:=SmallGroup(432,312);
// by ID

G=gap.SmallGroup(432,312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽