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G = C2×C4×He3order 216 = 23·33

Direct product of C2×C4 and He3

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

Aliases: C2×C4×He3, C62.4C6, C6.6C62, (C6×C12)⋊C3, (C3×C12)⋊4C6, (C3×C6)⋊3C12, C3.1(C6×C12), C6.4(C3×C12), C22.(C2×He3), C12.12(C3×C6), C326(C2×C12), (C2×C12).1C32, C2.1(C22×He3), (C22×He3).3C2, (C2×He3).16C22, (C3×C6).11(C2×C6), (C2×C6).10(C3×C6), SmallGroup(216,74)

Series: Derived Chief Lower central Upper central

C1C3 — C2×C4×He3
C1C3C6C3×C6C2×He3C4×He3 — C2×C4×He3
C1C3 — C2×C4×He3
C1C2×C12 — C2×C4×He3

Generators and relations for C2×C4×He3
 G = < a,b,c,d,e | a2=b4=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 152 in 88 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, C6, C6 [×2], C6 [×12], C2×C4, C32 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×4], C3×C6 [×12], C2×C12, C2×C12 [×4], He3, C3×C12 [×8], C62 [×4], C2×He3, C2×He3 [×2], C6×C12 [×4], C4×He3 [×2], C22×He3, C2×C4×He3
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, C32, C12 [×8], C2×C6 [×4], C3×C6 [×3], C2×C12 [×4], He3, C3×C12 [×2], C62, C2×He3 [×3], C6×C12, C4×He3 [×2], C22×He3, C2×C4×He3

Smallest permutation representation of C2×C4×He3
On 72 points
Generators in S72
(1 8)(2 5)(3 6)(4 7)(9 22)(10 23)(11 24)(12 21)(13 26)(14 27)(15 28)(16 25)(17 48)(18 45)(19 46)(20 47)(29 69)(30 70)(31 71)(32 72)(33 53)(34 54)(35 55)(36 56)(37 64)(38 61)(39 62)(40 63)(41 68)(42 65)(43 66)(44 67)(49 58)(50 59)(51 60)(52 57)
(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 42 71)(2 43 72)(3 44 69)(4 41 70)(5 66 32)(6 67 29)(7 68 30)(8 65 31)(9 25 36)(10 26 33)(11 27 34)(12 28 35)(13 53 23)(14 54 24)(15 55 21)(16 56 22)(17 57 38)(18 58 39)(19 59 40)(20 60 37)(45 49 62)(46 50 63)(47 51 64)(48 52 61)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 45 26)(6 46 27)(7 47 28)(8 48 25)(9 31 61)(10 32 62)(11 29 63)(12 30 64)(21 70 37)(22 71 38)(23 72 39)(24 69 40)(33 66 49)(34 67 50)(35 68 51)(36 65 52)(41 60 55)(42 57 56)(43 58 53)(44 59 54)
(1 42 38)(2 43 39)(3 44 40)(4 41 37)(5 66 62)(6 67 63)(7 68 64)(8 65 61)(9 48 52)(10 45 49)(11 46 50)(12 47 51)(13 53 72)(14 54 69)(15 55 70)(16 56 71)(17 57 22)(18 58 23)(19 59 24)(20 60 21)(25 36 31)(26 33 32)(27 34 29)(28 35 30)

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

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

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

C2×C4×He3 is a maximal subgroup of
He37M4(2)  C62.19D6  C62.20D6  C62.21D6  He38M4(2)  C62.29D6  C62.30D6  C62.31D6  C62.36D6  C62.47D6

88 conjugacy classes

class 1 2A2B2C3A3B3C···3J4A4B4C4D6A···6F6G···6AD12A···12H12I···12AN
order1222333···344446···66···612···1212···12
size1111113···311111···13···31···13···3

88 irreducible representations

dim111111113333
type+++
imageC1C2C2C3C4C6C6C12He3C2×He3C2×He3C4×He3
kernelC2×C4×He3C4×He3C22×He3C6×C12C2×He3C3×C12C62C3×C6C2×C4C4C22C2
# reps12184168322428

Matrix representation of C2×C4×He3 in GL4(𝔽13) generated by

12000
01200
00120
00012
,
1000
0800
0080
0008
,
1000
0010
0001
0100
,
1000
0300
0030
0003
,
3000
0090
0003
0100
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,0,0,1,0,9,0,0,0,0,3,0] >;

C2×C4×He3 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm He}_3
% in TeX

G:=Group("C2xC4xHe3");
// GroupNames label

G:=SmallGroup(216,74);
// by ID

G=gap.SmallGroup(216,74);
# by ID

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

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

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