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G = C2xC4x3- 1+2order 216 = 23·33

Direct product of C2xC4 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: C2xC4x3- 1+2, C36:4C6, C18:2C12, C62.5C6, C6.7C62, (C2xC36):C3, (C6xC12).C3, C9:3(C2xC12), C18.6(C2xC6), C6.5(C3xC12), C3.2(C6xC12), (C3xC6).5C12, (C3xC12).8C6, (C2xC18).3C6, C32.(C2xC12), C12.13(C3xC6), (C2xC12).2C32, C22.(C2x3- 1+2), C2.1(C22x3- 1+2), (C2x3- 1+2).6C22, (C22x3- 1+2).2C2, (C2xC6).11(C3xC6), (C3xC6).12(C2xC6), SmallGroup(216,75)

Series: Derived Chief Lower central Upper central

C1C3 — C2xC4x3- 1+2
C1C3C6C3xC6C2x3- 1+2C4x3- 1+2 — C2xC4x3- 1+2
C1C3 — C2xC4x3- 1+2
C1C2xC12 — C2xC4x3- 1+2

Generators and relations for C2xC4x3- 1+2
 G = < a,b,c,d | a2=b4=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 80 in 64 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2xC4, C9, C32, C12, C12, C2xC6, C2xC6, C18, C3xC6, C3xC6, C2xC12, C2xC12, 3- 1+2, C36, C2xC18, C3xC12, C62, C2x3- 1+2, C2x3- 1+2, C2xC36, C6xC12, C4x3- 1+2, C22x3- 1+2, C2xC4x3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C32, C12, C2xC6, C3xC6, C2xC12, 3- 1+2, C3xC12, C62, C2x3- 1+2, C6xC12, C4x3- 1+2, C22x3- 1+2, C2xC4x3- 1+2

Smallest permutation representation of C2xC4x3- 1+2
On 72 points
Generators in S72
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 34)(20 35)(21 36)(22 28)(23 29)(24 30)(25 31)(26 32)(27 33)(55 70)(56 71)(57 72)(58 64)(59 65)(60 66)(61 67)(62 68)(63 69)
(1 18 19 70)(2 10 20 71)(3 11 21 72)(4 12 22 64)(5 13 23 65)(6 14 24 66)(7 15 25 67)(8 16 26 68)(9 17 27 69)(28 58 40 46)(29 59 41 47)(30 60 42 48)(31 61 43 49)(32 62 44 50)(33 63 45 51)(34 55 37 52)(35 56 38 53)(36 57 39 54)
(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 8 5)(3 6 9)(10 16 13)(11 14 17)(20 26 23)(21 24 27)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(56 62 59)(57 60 63)(65 71 68)(66 69 72)

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

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

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

C2xC4x3- 1+2 is a maximal subgroup of   C36.C12  Dic9:C12  C36:C12  D18:C12  D36:6C6

88 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A···6F6G···6L9A···9F12A···12H12I···12P18A···18R36A···36X
order1222333344446···66···69···912···1212···1218···1836···36
size1111113311111···13···33···31···13···33···33···3

88 irreducible representations

dim1111111111113333
type+++
imageC1C2C2C3C3C4C6C6C6C6C12C123- 1+2C2x3- 1+2C2x3- 1+2C4x3- 1+2
kernelC2xC4x3- 1+2C4x3- 1+2C22x3- 1+2C2xC36C6xC12C2x3- 1+2C36C2xC18C3xC12C62C18C3xC6C2xC4C4C22C2
# reps121624126422482428

Matrix representation of C2xC4x3- 1+2 in GL4(F37) generated by

36000
0100
0010
0001
,
6000
03100
00310
00031
,
1000
0190
013610
00360
,
26000
0100
01100
011026
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[1,0,0,0,0,1,1,0,0,9,36,36,0,0,10,0],[26,0,0,0,0,1,1,11,0,0,10,0,0,0,0,26] >;

C2xC4x3- 1+2 in GAP, Magma, Sage, TeX

C_2\times C_4\times 3_-^{1+2}
% in TeX

G:=Group("C2xC4xES-(3,1)");
// GroupNames label

G:=SmallGroup(216,75);
// by ID

G=gap.SmallGroup(216,75);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,338,519]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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