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## G = C8×He3⋊C2order 432 = 24·33

### Direct product of C8 and He3⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C8×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — C4×He3⋊C2 — C8×He3⋊C2
 Lower central He3 — C8×He3⋊C2
 Upper central C1 — C24

Generators and relations for C8×He3⋊C2
G = < a,b,c,d,e | a8=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 377 in 121 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, He3⋊C2, C2×He3, C3×C3⋊C8, C3×C24, S3×C12, He33C4, C4×He3, C2×He3⋊C2, S3×C24, He34C8, C8×He3, C4×He3⋊C2, C8×He3⋊C2
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C3⋊S3, C4×S3, C2×C3⋊S3, S3×C8, He3⋊C2, C4×C3⋊S3, C2×He3⋊C2, C8×C3⋊S3, C4×He3⋊C2, C8×He3⋊C2

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 24A ··· 24H 24I ··· 24X 24Y ··· 24AF order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 9 9 1 1 6 6 6 6 1 1 9 9 1 1 6 6 6 6 9 9 9 9 1 1 1 1 9 9 9 9 1 1 1 1 6 ··· 6 9 9 9 9 1 ··· 1 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D6 C4×S3 S3×C8 He3⋊C2 C2×He3⋊C2 C4×He3⋊C2 C8×He3⋊C2 kernel C8×He3⋊C2 He3⋊4C8 C8×He3 C4×He3⋊C2 He3⋊3C4 C2×He3⋊C2 He3⋊C2 C3×C24 C3×C12 C3×C6 C32 C8 C4 C2 C1 # reps 1 1 1 1 2 2 8 4 4 8 16 4 4 8 16

Matrix representation of C8×He3⋊C2 in GL3(𝔽73) generated by

 51 0 0 0 51 0 0 0 51
,
 1 7 0 0 72 1 0 72 0
,
 8 0 0 0 8 0 0 0 8
,
 8 56 0 72 65 1 8 65 0
,
 72 0 0 0 0 72 0 72 0
G:=sub<GL(3,GF(73))| [51,0,0,0,51,0,0,0,51],[1,0,0,7,72,72,0,1,0],[8,0,0,0,8,0,0,0,8],[8,72,8,56,65,65,0,1,0],[72,0,0,0,0,72,0,72,0] >;

C8×He3⋊C2 in GAP, Magma, Sage, TeX

C_8\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C8xHe3:C2");
// GroupNames label

G:=SmallGroup(432,173);
// by ID

G=gap.SmallGroup(432,173);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,1124,4037,537]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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