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G = C12.40S32order 432 = 24·33

40th non-split extension by C12 of S32 acting via S32/C32=C22

metabelian, supersoluble, monomial

Aliases: C12.40S32, (C3×Dic6)⋊8S3, C338D48C2, Dic65(C3⋊S3), (C3×C12).143D6, C3315(C4○D4), C3312D45C2, C3⋊Dic3.50D6, C33(D6.6D6), (C3×Dic3).15D6, C31(C12.26D6), C3220(C4○D12), C327(Q83S3), (C32×Dic6)⋊12C2, (C32×C6).43C23, (C32×C12).45C22, (C32×Dic3).15C22, (C4×C3⋊S3)⋊7S3, C6.53(C2×S32), (C12×C3⋊S3)⋊6C2, C4.14(S3×C3⋊S3), C338(C2×C4)⋊5C2, C12.36(C2×C3⋊S3), (C2×C3⋊S3).43D6, C6.6(C22×C3⋊S3), Dic3.3(C2×C3⋊S3), (C6×C3⋊S3).52C22, (C3×C6).101(C22×S3), (C3×C3⋊Dic3).53C22, (C2×C33⋊C2).7C22, C2.10(C2×S3×C3⋊S3), SmallGroup(432,665)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C12.40S32
C1C3C32C33C32×C6C32×Dic3C338(C2×C4) — C12.40S32
C33C32×C6 — C12.40S32
C1C2C4

Generators and relations for C12.40S32
 G = < a,b,c,d,e,f | a3=b3=c3=d4=e2=f2=1, ab=ba, ac=ca, ad=da, eae=faf=a-1, bc=cb, bd=db, ebe=fbf=b-1, dcd-1=ece=fcf=c-1, ede=d-1, df=fd, fef=d2e >

Subgroups: 2104 in 304 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2 [×3], C3, C3 [×4], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×22], C6, C6 [×4], C6 [×5], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×4], C32 [×4], Dic3 [×2], Dic3 [×4], C12, C12 [×4], C12 [×13], D6 [×22], C2×C6, C4○D4, C3×S3 [×4], C3⋊S3 [×19], C3×C6, C3×C6 [×4], C3×C6 [×4], Dic6, C4×S3 [×14], D12 [×17], C3⋊D4 [×2], C2×C12, C3×Q8 [×4], C33, C3×Dic3 [×8], C3×Dic3 [×4], C3⋊Dic3, C3×C12, C3×C12 [×4], C3×C12 [×6], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×18], C4○D12, Q83S3 [×4], C3×C3⋊S3, C33⋊C2 [×2], C32×C6, C6.D6 [×8], C3⋊D12 [×8], C3×Dic6 [×4], S3×C12 [×4], C4×C3⋊S3, C4×C3⋊S3 [×2], C12⋊S3 [×11], Q8×C32, C32×Dic3 [×2], C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C2×C33⋊C2 [×2], D6.6D6 [×4], C12.26D6, C338(C2×C4) [×2], C338D4 [×2], C32×Dic6, C12×C3⋊S3, C3312D4, C12.40S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], C23, D6 [×15], C4○D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], C4○D12, Q83S3 [×4], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, D6.6D6 [×4], C12.26D6, C2×S3×C3⋊S3, C12.40S32

Smallest permutation representation of C12.40S32
On 72 points
Generators in S72
(1 37 11)(2 38 12)(3 39 9)(4 40 10)(5 62 28)(6 63 25)(7 64 26)(8 61 27)(13 31 60)(14 32 57)(15 29 58)(16 30 59)(17 42 50)(18 43 51)(19 44 52)(20 41 49)(21 66 45)(22 67 46)(23 68 47)(24 65 48)(33 70 55)(34 71 56)(35 72 53)(36 69 54)
(1 61 56)(2 62 53)(3 63 54)(4 64 55)(5 72 12)(6 69 9)(7 70 10)(8 71 11)(13 44 46)(14 41 47)(15 42 48)(16 43 45)(17 65 58)(18 66 59)(19 67 60)(20 68 57)(21 30 51)(22 31 52)(23 32 49)(24 29 50)(25 36 39)(26 33 40)(27 34 37)(28 35 38)
(1 8 34)(2 35 5)(3 6 36)(4 33 7)(9 25 54)(10 55 26)(11 27 56)(12 53 28)(13 19 22)(14 23 20)(15 17 24)(16 21 18)(29 42 65)(30 66 43)(31 44 67)(32 68 41)(37 61 71)(38 72 62)(39 63 69)(40 70 64)(45 51 59)(46 60 52)(47 49 57)(48 58 50)
(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)
(2 4)(5 33)(6 36)(7 35)(8 34)(9 39)(10 38)(11 37)(12 40)(13 58)(14 57)(15 60)(16 59)(17 46)(18 45)(19 48)(20 47)(21 51)(22 50)(23 49)(24 52)(25 69)(26 72)(27 71)(28 70)(29 31)(41 68)(42 67)(43 66)(44 65)(53 64)(54 63)(55 62)(56 61)
(1 29)(2 30)(3 31)(4 32)(5 66)(6 67)(7 68)(8 65)(9 60)(10 57)(11 58)(12 59)(13 39)(14 40)(15 37)(16 38)(17 71)(18 72)(19 69)(20 70)(21 62)(22 63)(23 64)(24 61)(25 46)(26 47)(27 48)(28 45)(33 41)(34 42)(35 43)(36 44)(49 55)(50 56)(51 53)(52 54)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J6K12A12B12C···12N12O···12V12W12X
order122223···33333444446···6666666121212···1212···121212
size111854542···24444266992···244441818224···412···121818

54 irreducible representations

dim111111222222224444
type++++++++++++++++
imageC1C2C2C2C2C2S3S3D6D6D6D6C4○D4C4○D12S32Q83S3C2×S32D6.6D6
kernelC12.40S32C338(C2×C4)C338D4C32×Dic6C12×C3⋊S3C3312D4C3×Dic6C4×C3⋊S3C3×Dic3C3⋊Dic3C3×C12C2×C3⋊S3C33C32C12C32C6C3
# reps122111418151244448

Matrix representation of C12.40S32 in GL8(𝔽13)

110000000
111000000
00100000
00010000
00001000
00000100
000000012
000000112
,
110000000
111000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000001200
000011200
00000010
00000001
,
10000000
01000000
00730000
00560000
000001200
000012000
00000010
00000001
,
10000000
112000000
00100000
004120000
00000100
00001000
00000001
00000010
,
10000000
112000000
004110000
00190000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(13))| [1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,11,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C12.40S32 in GAP, Magma, Sage, TeX

C_{12}._{40}S_3^2
% in TeX

G:=Group("C12.40S3^2");
// GroupNames label

G:=SmallGroup(432,665);
// by ID

G=gap.SmallGroup(432,665);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^4=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=f*a*f=a^-1,b*c=c*b,b*d=d*b,e*b*e=f*b*f=b^-1,d*c*d^-1=e*c*e=f*c*f=c^-1,e*d*e=d^-1,d*f=f*d,f*e*f=d^2*e>;
// generators/relations

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