Copied to
clipboard

G = C2×C6.S32  order 432 = 24·33

Direct product of C2 and C6.S32

direct product, non-abelian, supersoluble, monomial

Aliases: C2×C6.S32, C62.7D6, C3⋊Dic3⋊5D6, He3⋊3(C22×C4), C6.16(S3×Dic3), C32⋊(C22×Dic3), C32⋊C12⋊6C22, He3⋊3C4⋊5C22, (C2×He3).11C23, C22.9(C32⋊D6), (C22×He3).7C22, C6.85(C2×S32), (C3×C6)⋊1(C4×S3), (C2×C6).53S32, C3⋊S3⋊(C2×Dic3), (C3×C6)⋊(C2×Dic3), C32⋊2(S3×C2×C4), (C2×C3⋊S3).9D6, C3.2(C2×S3×Dic3), C32⋊C6⋊2(C2×C4), (C2×C32⋊C6)⋊3C4, (C2×He3)⋊2(C2×C4), (C2×C3⋊Dic3)⋊3S3, (C2×C3⋊S3)⋊2Dic3, C2.2(C2×C32⋊D6), (C2×C32⋊C12)⋊7C2, (C2×He3⋊3C4)⋊6C2, (C22×C3⋊S3).2S3, (C3×C6).11(C22×S3), (C22×C32⋊C6).3C2, (C2×C32⋊C6).9C22, SmallGroup(432,317)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — C2×C6.S32
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C6.S32 — C2×C6.S32
Lower central
He3 — C2×C6.S32
Upper central
C1 — C22

Generators and relations for C2×C6.S32
 G = < a,b,c,d,e | a2=b3=c3=d6=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, dbd-1=b-1c-1, dcd-1=ece-1=c-1, ede-1=d-1 >

Subgroups: 1019 in 205 conjugacy classes, 61 normal (21 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, S3×C2×C4, C22×Dic3, C32⋊C6, C2×He3, C2×He3, S3×Dic3, C6.D6, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C32⋊C12, He3⋊3C4, C2×C32⋊C6, C22×He3, C2×S3×Dic3, C2×C6.D6, C6.S32, C2×C32⋊C12, C2×He3⋊3C4, C22×C32⋊C6, C2×C6.S32
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, S32, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, C32⋊D6, C2×S3×Dic3, C6.S32, C2×C32⋊D6, C2×C6.S32

Smallest permutation representation of C2×C6.S32
►On 72 points
Generators in S72
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 66)(8 61)(9 62)(10 63)(11 64)(12 65)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

(1 26 22)(2 27 23)(4 19 29)(5 20 30)(8 71 14)(9 72 15)(11 17 68)(12 18 69)(31 41 48)(33 44 37)(34 45 38)(36 40 47)(50 64 57)(51 65 58)(53 60 61)(54 55 62)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A···4H6A6B6C6D···6I6J6K6L6M6N6O6P12A···12H
order1222222233334···46666···6666666612···12
size11119999266129···92226···61212121818181818···18

44 irreducible representations

dim1111112222222444666
type++++++++-+++-+++
imageC1C2C2C2C2C4S3S3D6Dic3D6D6C4×S3S32S3×Dic3C2×S32C32⋊D6C6.S32C2×C32⋊D6
kernelC2×C6.S32C6.S32C2×C32⋊C12C2×He3⋊3C4C22×C32⋊C6C2×C32⋊C6C2×C3⋊Dic3C22×C3⋊S3C3⋊Dic3C2×C3⋊S3C2×C3⋊S3C62C3×C6C2×C6C6C6C22C2C2
# reps1411181124224121242

Matrix representation of C2×C6.S32 ►in GL10(𝔽13)

12000000000
01200000000
00120000000
00012000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
0100000000
121200000000
0001000000
001212000000
0000010000
000012120000
0000001000
0000000100
000000001212
0000000010
,
1000000000
0100000000
0010000000
0001000000
0000010000
000012120000
0000000100
000000121200
0000000001
000000001212
,
0010000000
001212000000
120120000000
1111000000
00000012000
0000001100
00000000120
0000000011
00001200000
0000110000
,
1000000000
0100000000
120120000000
012012000000
0000800000
0000550000
0000000080
0000000055
0000008000
0000005500

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,12,1,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,12,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0] >;
 

C2×C6.S32 in GAP, Magma, Sage, TeX 

C_2\times C_6.S_3^2
 
% in TeX
 
G:=Group("C2xC6.S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,317);
 
// by ID
 
G=gap.SmallGroup(432,317);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^6=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁