Copied to
clipboard

G = C12⋊S32  order 432 = 24·33

2nd semidirect product of C12 and S32 acting via S32/C32=C22

metabelian, supersoluble, monomial

Aliases: C12⋊2S32, (S3×C6)⋊4D6, (C3×D12)⋊9S3, (C3×C12)⋊13D6, C33⋊15(C2×D4), D12⋊4(C3⋊S3), C12⋊S3⋊11S3, C32⋊11(S3×D4), C33⋊6D4⋊7C2, C33⋊C2⋊3D4, C3⋊2(D6⋊D6), (C32×D12)⋊13C2, (C32×C12)⋊5C22, C33⋊5C4⋊8C22, (C32×C6).51C23, C4⋊3(S3×C3⋊S3), C3⋊2(D4×C3⋊S3), C12⋊2(C2×C3⋊S3), C6.61(C2×S32), D6⋊2(C2×C3⋊S3), (C2×C3⋊S3)⋊15D6, (C6×C3⋊S3)⋊9C22, (S3×C3×C6)⋊12C22, (C4×C33⋊C2)⋊3C2, (C3×C12⋊S3)⋊11C2, C6.14(C22×C3⋊S3), (C3×C6).107(C22×S3), (C2×C33⋊C2).16C22, (C2×S3×C3⋊S3)⋊7C2, C2.17(C2×S3×C3⋊S3), SmallGroup(432,673)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C12⋊S32
 G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, cac=a-1, ad=da, eae=a7, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2624 in 388 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, D12, C3⋊D4, C3×D4, C22×S3, C33, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, D6⋊S3, C3×D12, C3×D12, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C2×S32, C22×C3⋊S3, C33⋊5C4, C32×C12, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, D6⋊D6, D4×C3⋊S3, C33⋊6D4, C32×D12, C3×C12⋊S3, C4×C33⋊C2, C2×S3×C3⋊S3, C12⋊S32
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, S3×D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6⋊D6, D4×C3⋊S3, C2×S3×C3⋊S3, C12⋊S32

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

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6E6F6G6H6I6J···6Q6R6S12A···12M
order122222223···33333446···666666···66612···12
size1166181827272···244442542···2444412···1236364···4

51 irreducible representations

dim1111112222224444
type+++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6S32S3×D4C2×S32D6⋊D6
kernelC12⋊S32C33⋊6D4C32×D12C3×C12⋊S3C4×C33⋊C2C2×S3×C3⋊S3C3×D12C12⋊S3C33⋊C2C3×C12S3×C6C2×C3⋊S3C12C32C6C3
# reps1211124125824548

Matrix representation of C12⋊S32 ►in GL8(𝔽13)

012000000
112000000
001230000
00810000
000012000
000001200
00000010
00000001
,
121000000
120000000
00100000
00010000
000012100
000012000
00000010
00000001
,
01000000
10000000
00100000
005120000
000001200
000012000
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000001212
00000010
,
10000000
01000000
001200000
00810000
000012000
000001200
00000010
0000001212

G:=sub<GL(8,GF(13))| [0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;
 

C12⋊S32 in GAP, Magma, Sage, TeX 

C_{12}\rtimes S_3^2
 
% in TeX
 
G:=Group("C12:S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,673);
 
// by ID
 
G=gap.SmallGroup(432,673);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,e*a*e=a^7,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁