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G = He3⋊4(C2×C8)  order 432 = 24·33

2nd semidirect product of He3 and C2×C8 acting via C2×C8/C4=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊4(C2×C8), C32⋊2(S3×C8), C12.89(S32), He3⋊3C8⋊7C2, He3⋊C2⋊3C8, C32⋊4C8⋊6S3, (C3×C12).37D6, He3⋊3C4.4C4, C4.15(C32⋊D6), C6.12(C6.D6), (C4×He3).29C22, C3.2(C12.29D6), (C3×C6).3(C4×S3), (C2×He3).10(C2×C4), (C2×He3⋊C2).2C4, (C4×He3⋊C2).4C2, C2.1(He3⋊(C2×C4)), SmallGroup(432,81)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — He3 — He3⋊4(C2×C8)
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊3C8 — He3⋊4(C2×C8)
Lower central
He3 — He3⋊4(C2×C8)
Upper central
C1 — C4

Generators and relations for He3⋊4(C2×C8)
 G = < a,b,c,d,e | a3=b3=c3=d2=e8=1, ab=ba, cac-1=ab-1, dad=a-1, ae=ea, bc=cb, bd=db, ebe-1=b-1, dcd=ece-1=c-1, de=ed >

Subgroups: 403 in 93 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4, C22, S3 [×6], C6, C6 [×5], C8 [×2], C2×C4, C32 [×2], C32, Dic3 [×3], C12, C12 [×4], D6 [×3], C2×C6, C2×C8, C3×S3 [×6], C3×C6 [×2], C3×C6, C3⋊C8 [×4], C24 [×2], C4×S3 [×3], C2×C12, He3, C3×Dic3 [×3], C3×C12 [×2], C3×C12, S3×C6 [×3], S3×C8 [×2], C2×C3⋊C8, He3⋊C2 [×2], C2×He3, C3×C3⋊C8 [×2], C32⋊4C8 [×2], S3×C12 [×3], He3⋊3C4, C4×He3, C2×He3⋊C2, S3×C3⋊C8 [×2], He3⋊3C8 [×2], C4×He3⋊C2, He3⋊4(C2×C8)
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D6 [×2], C2×C8, C4×S3 [×2], S32, S3×C8 [×2], C6.D6, C32⋊D6, C12.29D6, He3⋊(C2×C4), He3⋊4(C2×C8)

Smallest permutation representation of He3⋊4(C2×C8)
►On 72 points
Generators in S72
(1 68 56)(2 69 49)(3 70 50)(4 71 51)(5 72 52)(6 65 53)(7 66 54)(8 67 55)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 41 27)(18 42 28)(19 43 29)(20 44 30)(21 45 31)(22 46 32)(23 47 25)(24 48 26)

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

(17 39 56)(18 49 40)(19 33 50)(20 51 34)(21 35 52)(22 53 36)(23 37 54)(24 55 38)(25 66 61)(26 62 67)(27 68 63)(28 64 69)(29 70 57)(30 58 71)(31 72 59)(32 60 65)

(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)

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

44 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F8A···8H12A12B12C12D12E12F12G12H12I12J24A···24H
order1222333344446666668···81212121212121212121224···24
size11992661211992661218189···92266661212181818···18

44 irreducible representations

dim1111112222444666
type++++++++-
imageC1C2C2C4C4C8S3D6C4×S3S3×C8S32C6.D6C12.29D6C32⋊D6He3⋊(C2×C4)He3⋊4(C2×C8)
kernelHe3⋊4(C2×C8)He3⋊3C8C4×He3⋊C2He3⋊3C4C2×He3⋊C2He3⋊C2C32⋊4C8C3×C12C3×C6C32C12C6C3C4C2C1
# reps1212282248112224

Matrix representation of He3⋊4(C2×C8) ►in GL10(𝔽73)

0010000000
0001000000
720720000000
072072000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
1000000000
0100000000
0010000000
0001000000
00007210000
00007200000
00000072100
00000072000
00000000721
00000000720
,
07200000000
17200000000
00072000000
00172000000
0000100000
0000010000
00000007200
00000017200
00000000721
00000000720
,
0100000000
1000000000
072072000000
720720000000
0000100000
0000010000
0000000010
0000000001
0000001000
0000000100
,
05100000000
51000000000
00051000000
00510000000
000043430000
000013300000
000000434300
000000133000
000000004343
000000001330

G:=sub<GL(10,GF(73))| [0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[0,1,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,51,0,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,43,30,0,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,43,30,0,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,43,30] >;
 

He3⋊4(C2×C8) in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_4(C_2\times C_8)
 
% in TeX
 
G:=Group("He3:4(C2xC8)");
 
// GroupNames label
 
G:=SmallGroup(432,81);
 
// by ID
 
G=gap.SmallGroup(432,81);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,36,58,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^2=e^8=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d=e*c*e^-1=c^-1,d*e=e*d>;
 
// generators/relations
 

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