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G = C12.73S32  order 432 = 24·33

30th non-split extension by C12 of S32 acting via S32/C3xS3=C2

metabelian, supersoluble, monomial

Aliases: C12.73S32, (S3xC12):1S3, (S3xC6).40D6, (C3xC12).170D6, C33:16(C4oD4), C33:8D4:11C2, C33:7D4:11C2, C33:6D4:11C2, C3:Dic3.45D6, C33:4Q8:11C2, C3:4(D6.D6), (C3xDic3).35D6, C3:2(C12.59D6), C32:10(C4oD12), (C32xC6).45C23, (C32xC12).73C22, C33:5C4.16C22, (C32xDic3).22C22, (S3xC3xC12):1C2, C6.55(C2xS32), (C12xC3:S3):1C2, (C4xC3:S3):11S3, C4.28(S3xC3:S3), D6.5(C2xC3:S3), (C4xS3):4(C3:S3), C12.43(C2xC3:S3), (C2xC3:S3).44D6, C6.8(C22xC3:S3), (C4xC33:C2):7C2, (S3xC3xC6).24C22, Dic3.8(C2xC3:S3), (C6xC3:S3).53C22, (C3xC6).103(C22xS3), (C3xC3:Dic3).43C22, (C2xC33:C2).14C22, C2.12(C2xS3xC3:S3), SmallGroup(432,667)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — C12.73S32
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — C33:6D4 — C12.73S32
Lower central
C33 — C32xC6 — C12.73S32
Upper central
C1 — C4

Generators and relations for C12.73S32
 G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a5, cbc=b-1, bd=db, be=eb, cd=dc, ece=a6c, ede=d-1 >

Subgroups: 1784 in 304 conjugacy classes, 68 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2xC4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, C4xS3, D12, C3:D4, C2xC12, C33, C3xDic3, C3xDic3, C3:Dic3, C3:Dic3, C3xC12, C3xC12, C3xC12, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, C4oD12, S3xC32, C3xC3:S3, C33:C2, C32xC6, D6:S3, C3:D12, C32:2Q8, S3xC12, S3xC12, C32:4Q8, C4xC3:S3, C4xC3:S3, C12:S3, C32:7D4, C6xC12, C32xDic3, C3xC3:Dic3, C33:5C4, C32xC12, S3xC3xC6, C6xC3:S3, C2xC33:C2, D6.D6, C12.59D6, C33:6D4, C33:7D4, C33:8D4, C33:4Q8, S3xC3xC12, C12xC3:S3, C4xC33:C2, C12.73S32
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C3:S3, C22xS3, S32, C2xC3:S3, C4oD12, C2xS32, C22xC3:S3, S3xC3:S3, D6.D6, C12.59D6, C2xS3xC3:S3, C12.73S32

Smallest permutation representation of C12.73S32
►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 37 17)(2 38 18)(3 39 19)(4 40 20)(5 41 21)(6 42 22)(7 43 23)(8 44 24)(9 45 13)(10 46 14)(11 47 15)(12 48 16)(25 67 49)(26 68 50)(27 69 51)(28 70 52)(29 71 53)(30 72 54)(31 61 55)(32 62 56)(33 63 57)(34 64 58)(35 65 59)(36 66 60)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J···6Q6R6S12A···12J12K···12R12S···12Z12AA12AB
order122223···33333444446···666666···66612···1212···1212···121212
size11618542···2444411618542···244446···618182···24···46···61818

66 irreducible representations

dim11111111222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D6D6D6D6D6C4oD4C4oD12S32C2xS32D6.D6
kernelC12.73S32C33:6D4C33:7D4C33:8D4C33:4Q8S3xC3xC12C12xC3:S3C4xC33:C2S3xC12C4xC3:S3C3xDic3C3:Dic3C3xC12S3xC6C2xC3:S3C33C32C12C6C3
# reps111111114141541220448

Matrix representation of C12.73S32 ►in GL8(F13)

50000000
05000000
001120000
00100000
000012000
000001200
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000012
000000112
,
72000000
26000000
001200000
000120000
000012000
000001200
00000001
00000010
,
10000000
01000000
001210000
001200000
000012100
000012000
00000010
00000001
,
43000000
89000000
00010000
00100000
00000100
00001000
00000010
00000001

G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[7,2,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,8,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C12.73S32 in GAP, Magma, Sage, TeX 

C_{12}._{73}S_3^2
 
% in TeX
 
G:=Group("C12.73S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,667);
 
// by ID
 
G=gap.SmallGroup(432,667);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c=b^-1,b*d=d*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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