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G = C2×D4×C3⋊S3  order 288 = 25·32

Direct product of C2, D4 and C3⋊S3

direct product, metabelian, supersoluble, monomial, rational

Aliases: C2×D4×C3⋊S3, C62⋊5C23, C6⋊4(S3×D4), (C6×D4)⋊5S3, (C2×C12)⋊7D6, (C3×D4)⋊16D6, C12⋊3(C22×S3), (C3×C12)⋊5C23, (C22×C6)⋊10D6, (C6×C12)⋊13C22, C6.58(S3×C23), (C3×C6).57C24, C3⋊Dic3⋊7C23, C32⋊12(C22×D4), (C2×C62)⋊11C22, C12⋊S3⋊25C22, (D4×C32)⋊23C22, C32⋊7D4⋊11C22, C3⋊5(C2×S3×D4), (D4×C3×C6)⋊12C2, (C3×C6)⋊11(C2×D4), C23⋊4(C2×C3⋊S3), C4⋊1(C22×C3⋊S3), (C2×C3⋊S3)⋊7C23, (C23×C3⋊S3)⋊7C2, (C2×C6)⋊6(C22×S3), C2.6(C23×C3⋊S3), (C4×C3⋊S3)⋊14C22, (C2×C12⋊S3)⋊20C2, C22⋊2(C22×C3⋊S3), (C2×C32⋊7D4)⋊18C2, (C22×C3⋊S3)⋊16C22, (C2×C3⋊Dic3)⋊26C22, (C2×C4×C3⋊S3)⋊7C2, (C2×C4)⋊6(C2×C3⋊S3), SmallGroup(288,1007)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×D4×C3⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C23×C3⋊S3 — C2×D4×C3⋊S3
Lower central
C32 — C3×C6 — C2×D4×C3⋊S3
Upper central
C1 — C22 — C2×D4

Generators and relations for C2×D4×C3⋊S3
 G = < a,b,c,d,e,f | a2=b4=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 2916 in 708 conjugacy classes, 173 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C24, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, D4×C32, C22×C3⋊S3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C2×S3×D4, C2×C4×C3⋊S3, C2×C12⋊S3, D4×C3⋊S3, C2×C32⋊7D4, D4×C3×C6, C23×C3⋊S3, C2×D4×C3⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊S3, C22×S3, C22×D4, C2×C3⋊S3, S3×D4, S3×C23, C22×C3⋊S3, C2×S3×D4, D4×C3⋊S3, C23×C3⋊S3, C2×D4×C3⋊S3

Smallest permutation representation of C2×D4×C3⋊S3
►On 72 points
Generators in S72
(1 27)(2 28)(3 25)(4 26)(5 12)(6 9)(7 10)(8 11)(13 23)(14 24)(15 21)(16 22)(17 46)(18 47)(19 48)(20 45)(29 68)(30 65)(31 66)(32 67)(33 55)(34 56)(35 53)(36 54)(37 57)(38 58)(39 59)(40 60)(41 71)(42 72)(43 69)(44 70)(49 62)(50 63)(51 64)(52 61)

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O3A3B3C3D4A4B4C4D6A···6L6M···6AB12A···12H
order1222222222222222333344446···66···612···12
size1111222299991818181822222218182···24···44···4

60 irreducible representations

dim1111111222224
type+++++++++++++
imageC1C2C2C2C2C2C2S3D4D6D6D6S3×D4
kernelC2×D4×C3⋊S3C2×C4×C3⋊S3C2×C12⋊S3D4×C3⋊S3C2×C32⋊7D4D4×C3×C6C23×C3⋊S3C6×D4C2×C3⋊S3C2×C12C3×D4C22×C6C6
# reps11182124441688

Matrix representation of C2×D4×C3⋊S3 ►in GL6(ℤ)

-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
001000
000100
000001
0000-10
,
-100000
0-10000
00-1000
000-100
000001
000010
,
010000
-1-10000
000100
00-1-100
000010
000001
,
-1-10000
100000
000100
00-1-100
000010
000001
,
-1-10000
010000
000-100
00-1000
000010
000001

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C2×D4×C3⋊S3 in GAP, Magma, Sage, TeX 

C_2\times D_4\times C_3\rtimes S_3
 
% in TeX
 
G:=Group("C2xD4xC3:S3");
 
// GroupNames label
 
G:=SmallGroup(288,1007);
 
// by ID
 
G=gap.SmallGroup(288,1007);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,185,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^4=c^2=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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