Copied to
clipboard

G = C12.39S32  order 432 = 24·33

39th non-split extension by C12 of S32 acting via S32/C32=C22

metabelian, supersoluble, monomial

Aliases: C12.39S32, (C3×Dic6)⋊7S3, C12⋊S3⋊10S3, C33⋊8D4⋊7C2, Dic6⋊3(C3⋊S3), (C3×C12).142D6, C33⋊14(C4○D4), C3⋊3(D12⋊S3), (C3×Dic3).14D6, C3⋊2(C12.26D6), C32⋊6(Q8⋊3S3), (C32×Dic6)⋊11C2, (C32×C6).42C23, C32⋊20(D4⋊2S3), (C32×C12).44C22, C33⋊5C4.14C22, (C32×Dic3).14C22, C6.52(C2×S32), C4.20(S3×C3⋊S3), C12.23(C2×C3⋊S3), (Dic3×C3⋊S3)⋊3C2, (C2×C3⋊S3).34D6, (C3×C12⋊S3)⋊8C2, C6.5(C22×C3⋊S3), (C4×C33⋊C2)⋊2C2, Dic3.2(C2×C3⋊S3), (C6×C3⋊S3).26C22, (C3×C6).100(C22×S3), (C2×C33⋊C2).12C22, C2.9(C2×S3×C3⋊S3), SmallGroup(432,664)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C12.39S32
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — Dic3×C3⋊S3 — C12.39S32
Lower central
C33 — C32×C6 — C12.39S32
Upper central
C1 — C2 — C4

Generators and relations for C12.39S32
 G = < a,b,c,d,e | a3=b12=d3=e2=1, c2=b6, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=b-1, bd=db, ebe=b7, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1824 in 304 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, D4⋊2S3, Q8⋊3S3, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, C32×Dic3, C33⋊5C4, C32×C12, C6×C3⋊S3, C2×C33⋊C2, D12⋊S3, C12.26D6, Dic3×C3⋊S3, C33⋊8D4, C32×Dic6, C3×C12⋊S3, C4×C33⋊C2, C12.39S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, D4⋊2S3, Q8⋊3S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D12⋊S3, C12.26D6, C2×S3×C3⋊S3, C12.39S32

Smallest permutation representation of C12.39S32
►On 72 points
Generators in S72
(1 30 40)(2 31 41)(3 32 42)(4 33 43)(5 34 44)(6 35 45)(7 36 46)(8 25 47)(9 26 48)(10 27 37)(11 28 38)(12 29 39)(13 66 54)(14 67 55)(15 68 56)(16 69 57)(17 70 58)(18 71 59)(19 72 60)(20 61 49)(21 62 50)(22 63 51)(23 64 52)(24 65 53)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J6K12A···12M12N···12U
order122223···33333444446···666666612···1212···12
size111818542···2444426627272···2444436364···412···12

51 irreducible representations

dim11111122222244444
type++++++++++++-++
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4S32D4⋊2S3Q8⋊3S3C2×S32D12⋊S3
kernelC12.39S32Dic3×C3⋊S3C33⋊8D4C32×Dic6C3×C12⋊S3C4×C33⋊C2C3×Dic6C12⋊S3C3×Dic3C3×C12C2×C3⋊S3C33C12C32C32C6C3
# reps12211141852241448

Matrix representation of C12.39S32 ►in GL8(𝔽13)

10000000
01000000
001210000
001200000
00001000
00000100
00000010
00000001
,
80000000
05000000
00100000
00010000
000012000
000001200
000000121
000000120
,
05000000
50000000
00100000
00010000
000012000
000001200
00000001
00000010
,
10000000
01000000
000120000
001120000
000012100
000012000
00000010
00000001
,
01000000
10000000
00010000
00100000
00000100
00001000
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C12.39S32 in GAP, Magma, Sage, TeX 

C_{12}._{39}S_3^2
 
% in TeX
 
G:=Group("C12.39S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,664);
 
// by ID
 
G=gap.SmallGroup(432,664);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,135,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^12=d^3=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^7,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁