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G = C2×C6.S4order 288 = 25·32

Direct product of C2 and C6.S4

direct product, non-abelian, soluble, monomial

Aliases: C2×C6.S4, C24.D9, C23⋊Dic9, C23.4D18, C6.25(C2×S4), (C2×C6).10S4, C22⋊(C2×Dic9), C6.4(A4⋊C4), (C23×C6).2S3, (C22×C6).16D6, C22.6(C3.S4), (C22×C6).3Dic3, C3.(C2×A4⋊C4), (C2×C3.A4)⋊C4, C3.A42(C2×C4), C2.2(C2×C3.S4), (C2×C6).(C2×Dic3), (C22×C3.A4).C2, (C2×C3.A4).4C22, SmallGroup(288,341)

Series: Derived Chief Lower central Upper central

C1C22C3.A4 — C2×C6.S4
C1C22C2×C6C3.A4C2×C3.A4C6.S4 — C2×C6.S4
C3.A4 — C2×C6.S4
C1C22

Generators and relations for C2×C6.S4
 G = < a,b,c,d,e,f | a2=b6=c2=d2=1, e3=b2, f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=b4e2 >

Subgroups: 498 in 110 conjugacy classes, 28 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C22 [×2], C22 [×10], C6, C6 [×2], C6 [×4], C2×C4 [×8], C23, C23 [×2], C23 [×4], C9, Dic3 [×4], C2×C6 [×2], C2×C6 [×10], C22⋊C4 [×4], C22×C4 [×2], C24, C18 [×3], C2×Dic3 [×8], C22×C6, C22×C6 [×2], C22×C6 [×4], C2×C22⋊C4, Dic9 [×2], C3.A4, C2×C18, C6.D4 [×4], C22×Dic3 [×2], C23×C6, C2×Dic9, C2×C3.A4, C2×C3.A4 [×2], C2×C6.D4, C6.S4 [×2], C22×C3.A4, C2×C6.S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, D9, C2×Dic3, S4, Dic9 [×2], D18, A4⋊C4 [×2], C2×S4, C2×Dic9, C3.S4, C2×A4⋊C4, C6.S4 [×2], C2×C3.S4, C2×C6.S4

Smallest permutation representation of C2×C6.S4
On 72 points
Generators in S72
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 64)(9 65)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 51)(20 52)(21 53)(22 54)(23 46)(24 47)(25 48)(26 49)(27 50)(28 59)(29 60)(30 61)(31 62)(32 63)(33 55)(34 56)(35 57)(36 58)
(1 62 4 56 7 59)(2 63 5 57 8 60)(3 55 6 58 9 61)(10 19 13 22 16 25)(11 20 14 23 17 26)(12 21 15 24 18 27)(28 66 31 69 34 72)(29 67 32 70 35 64)(30 68 33 71 36 65)(37 51 40 54 43 48)(38 52 41 46 44 49)(39 53 42 47 45 50)
(1 42)(2 51)(3 58)(4 45)(5 54)(6 61)(7 39)(8 48)(9 55)(10 29)(11 23)(12 72)(13 32)(14 26)(15 66)(16 35)(17 20)(18 69)(19 67)(21 28)(22 70)(24 31)(25 64)(27 34)(30 71)(33 65)(36 68)(37 60)(38 46)(40 63)(41 49)(43 57)(44 52)(47 62)(50 56)(53 59)
(1 56)(2 43)(3 52)(4 59)(5 37)(6 46)(7 62)(8 40)(9 49)(10 70)(11 30)(12 24)(13 64)(14 33)(15 27)(16 67)(17 36)(18 21)(19 35)(20 68)(22 29)(23 71)(25 32)(26 65)(28 69)(31 72)(34 66)(38 61)(39 47)(41 55)(42 50)(44 58)(45 53)(48 63)(51 57)(54 60)
(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 50 56 42)(2 49 57 41)(3 48 58 40)(4 47 59 39)(5 46 60 38)(6 54 61 37)(7 53 62 45)(8 52 63 44)(9 51 55 43)(10 71 22 30)(11 70 23 29)(12 69 24 28)(13 68 25 36)(14 67 26 35)(15 66 27 34)(16 65 19 33)(17 64 20 32)(18 72 21 31)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4H6A6B6C6D6E6F6G9A9B9C18A···18I
order1222222234···4666666699918···18
size11113333218···1822266668888···8

36 irreducible representations

dim1111222222333666
type++++-++-++++-+
imageC1C2C2C4S3Dic3D6D9Dic9D18S4A4⋊C4C2×S4C3.S4C6.S4C2×C3.S4
kernelC2×C6.S4C6.S4C22×C3.A4C2×C3.A4C23×C6C22×C6C22×C6C24C23C23C2×C6C6C6C22C2C2
# reps1214121363242121

Matrix representation of C2×C6.S4 in GL5(𝔽37)

360000
036000
00100
00010
00001
,
136000
10000
003600
000360
000036
,
10000
01000
00010
00100
000036
,
10000
01000
003600
000360
003161
,
206000
3126000
00001
0063136
0016156
,
2936000
288000
00010
003600
0022216

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,1,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,31,0,0,0,36,6,0,0,0,0,1],[20,31,0,0,0,6,26,0,0,0,0,0,0,6,16,0,0,0,31,15,0,0,1,36,6],[29,28,0,0,0,36,8,0,0,0,0,0,0,36,22,0,0,1,0,21,0,0,0,0,6] >;

C2×C6.S4 in GAP, Magma, Sage, TeX

C_2\times C_6.S_4
% in TeX

G:=Group("C2xC6.S4");
// GroupNames label

G:=SmallGroup(288,341);
// by ID

G=gap.SmallGroup(288,341);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,1123,192,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=d^2=1,e^3=b^2,f^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b^-1,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=b^4*e^2>;
// generators/relations

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