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G = C12.91S32  order 432 = 24·33

11st non-split extension by C12 of S32 acting via S32/C3:S3=C2

non-abelian, supersoluble, monomial

Aliases: C12.91S32, He3:3(C4oD4), He3:2D4:6C2, He3:3D4:7C2, He3:2Q8:6C2, (C3xC12).39D6, C3:Dic3.7D6, C32:2(C4oD12), C4.17(C32:D6), (C2xHe3).4C23, C3.2(D6.D6), C32:C12.6C22, (C4xHe3).31C22, He3:3C4.11C22, (C4xC3:S3):4S3, C6.78(C2xS32), (C2xC3:S3).6D6, (C4xC32:C6):1C2, C2.7(C2xC32:D6), (C4xHe3:C2):1C2, (C3xC6).4(C22xS3), (C2xC32:C6).6C22, (C2xHe3:C2).12C22, SmallGroup(432,297)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2xHe3 — C12.91S32
Chief series
C1 — C3 — C32 — He3 — C2xHe3 — C2xC32:C6 — He3:2D4 — C12.91S32
Lower central
He3 — C2xHe3 — C12.91S32
Upper central
C1 — C4

Generators and relations for C12.91S32
 G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, cac=eae=a5, ad=da, cbc=b-1, dbd-1=a4b, be=eb, cd=dc, ece=a6c, ede=d-1 >

Subgroups: 927 in 156 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, He3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, C4oD12, C32:C6, He3:C2, C2xHe3, D6:S3, C3:D12, C32:2Q8, S3xC12, C4xC3:S3, C32:C12, He3:3C4, C4xHe3, C2xC32:C6, C2xHe3:C2, D6.D6, He3:2Q8, He3:2D4, He3:3D4, C4xC32:C6, C4xHe3:C2, C12.91S32
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, C4oD12, C2xS32, C32:D6, D6.D6, C2xC32:D6, C12.91S32

Smallest permutation representation of C12.91S32
►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 55 18)(2 56 19)(3 57 20)(4 58 21)(5 59 22)(6 60 23)(7 49 24)(8 50 13)(9 51 14)(10 52 15)(11 53 16)(12 54 17)(25 41 64)(26 42 65)(27 43 66)(28 44 67)(29 45 68)(30 46 69)(31 47 70)(32 48 71)(33 37 72)(34 38 61)(35 39 62)(36 40 63)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E···6J12A12B12C12D12E12F12G12H12I···12N
order1222233334444466666···6121212121212121212···12
size1118181826612111818182661218···18226666121218···18

38 irreducible representations

dim111111222222444666
type++++++++++++++
imageC1C2C2C2C2C2S3D6D6D6C4oD4C4oD12S32C2xS32D6.D6C32:D6C2xC32:D6C12.91S32
kernelC12.91S32He3:2Q8He3:2D4He3:3D4C4xC32:C6C4xHe3:C2C4xC3:S3C3:Dic3C3xC12C2xC3:S3He3C32C12C6C3C4C2C1
# reps111221222228112224

Matrix representation of C12.91S32 ►in GL10(F13)

8000000000
0800000000
0080000000
0008000000
00000120000
0000110000
00000001200
0000001100
00000000012
0000000011
,
121200000000
1000000000
0001000000
001212000000
000000001212
0000000010
0000010000
000012120000
0000001000
0000000100
,
0030000000
0003000000
9000000000
0900000000
0000300303
000010103030
0000033003
000030101030
0000030330
000030301010
,
0100000000
121200000000
0001000000
001212000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
00110000000
0022000000
6000000000
7700000000
0000100010010
000033100100
0000010010100
000010010033
0000010100010
000010033100

G:=sub<GL(10,GF(13))| [8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0],[0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,10,0,3,0,3,0,0,0,0,0,10,3,0,3,0,0,0,0,0,0,3,3,10,0,3,0,0,0,0,3,0,0,10,3,0,0,0,0,0,0,3,0,3,3,10,0,0,0,0,3,0,3,0,0,10],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,0,6,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,11,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,10,3,0,10,0,10,0,0,0,0,0,3,10,0,10,0,0,0,0,0,0,10,0,10,10,3,0,0,0,0,10,0,10,0,0,3,0,0,0,0,0,10,10,3,0,10,0,0,0,0,10,0,0,3,10,0] >;
 

C12.91S32 in GAP, Magma, Sage, TeX 

C_{12}._{91}S_3^2
 
% in TeX
 
G:=Group("C12.91S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,297);
 
// by ID
 
G=gap.SmallGroup(432,297);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,571,4037,537,14118,7069]);
 
// 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=e*a*e=a^5,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a^4*b,b*e=e*b,c*d=d*c,e*c*e=a^6*c,e*d*e=d^-1>;
 
// generators/relations
 

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