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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(C4○D4), He3⋊2D4⋊6C2, He3⋊3D4⋊7C2, He3⋊2Q8⋊6C2, (C3×C12).39D6, C3⋊Dic3.7D6, C32⋊2(C4○D12), C4.17(C32⋊D6), (C2×He3).4C23, C3.2(D6.D6), C32⋊C12.6C22, (C4×He3).31C22, He3⋊3C4.11C22, (C4×C3⋊S3)⋊4S3, C6.78(C2×S32), (C2×C3⋊S3).6D6, (C4×C32⋊C6)⋊1C2, C2.7(C2×C32⋊D6), (C4×He3⋊C2)⋊1C2, (C3×C6).4(C22×S3), (C2×C32⋊C6).6C22, (C2×He3⋊C2).12C22, SmallGroup(432,297)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C12.91S32
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊2D4 — C12.91S32
Lower central
He3 — C2×He3 — 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, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, C32⋊C6, He3⋊C2, C2×He3, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, C32⋊C12, He3⋊3C4, C4×He3, C2×C32⋊C6, C2×He3⋊C2, D6.D6, He3⋊2Q8, He3⋊2D4, He3⋊3D4, C4×C32⋊C6, C4×He3⋊C2, C12.91S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, C2×S32, C32⋊D6, D6.D6, C2×C32⋊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++++++++++++++
imageC1C2C2C2C2C2S3D6D6D6C4○D4C4○D12S32C2×S32D6.D6C32⋊D6C2×C32⋊D6C12.91S32
kernelC12.91S32He3⋊2Q8He3⋊2D4He3⋊3D4C4×C32⋊C6C4×He3⋊C2C4×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊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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