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G = C12.91(S32)  order 432 = 24·33

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

non-abelian, supersoluble, monomial

Aliases: C12.91(S32), He3⋊3(C4○D4), He3⋊3D4⋊7C2, He3⋊2D4⋊6C2, 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 — C12.91(S32)
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊2D4 — C12.91(S32)
Lower central
He3 — C2×He3 — C12.91(S32)
Upper central
C1 — C4

Generators and relations for C12.91(S32)
 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 [×3], C3, C3 [×3], C4, C4 [×3], C22 [×3], S3 [×8], C6, C6 [×6], C2×C4 [×3], D4 [×3], Q8, C32 [×2], C32, Dic3 [×7], C12, C12 [×6], D6 [×7], C2×C6 [×3], C4○D4, C3×S3 [×6], C3⋊S3 [×2], C3×C6 [×2], C3×C6, Dic6 [×3], C4×S3 [×7], D12 [×3], C3⋊D4 [×6], C2×C12 [×3], He3, C3×Dic3 [×5], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, S3×C6 [×5], C2×C3⋊S3 [×2], C4○D12 [×3], C32⋊C6 [×2], He3⋊C2, C2×He3, D6⋊S3 [×2], C3⋊D12 [×4], C32⋊2Q8 [×2], S3×C12 [×5], C4×C3⋊S3 [×2], C32⋊C12 [×2], He3⋊3C4, C4×He3, C2×C32⋊C6 [×2], C2×He3⋊C2, D6.D6 [×2], He3⋊2Q8, He3⋊2D4, He3⋊3D4 [×2], C4×C32⋊C6 [×2], C4×He3⋊C2, C12.91(S32)
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, C4○D12 [×2], C2×S32, C32⋊D6, D6.D6, C2×C32⋊D6, C12.91(S32)

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

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

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

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

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⋊D6C12.91(S32)
kernelC12.91(S32)He3⋊2Q8He3⋊2D4He3⋊3D4C4×C32⋊C6C4×He3⋊C2C4×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊S3He3C32C12C6C3C4C2C1
# reps111221222228112224

Matrix representation of C12.91(S32) ►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] >;
 

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

C_{12}._{91}(S_3^2)
 
% in TeX
 
G:=Group("C12.91(S3^2)");
 
// 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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