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G = C8×He3order 216 = 23·33

Direct product of C8 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C8×He3, C324C24, C24.1C32, (C3×C24)⋊C3, C2.(C4×He3), (C3×C12).6C6, (C3×C6).3C12, C6.2(C3×C12), C3.1(C3×C24), C4.2(C2×He3), C12.10(C3×C6), (C4×He3).6C2, (C2×He3).4C4, SmallGroup(216,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C8×He3
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×He3 — C8×He3
 Lower central C1 — C3 — C8×He3
 Upper central C1 — C24 — C8×He3

Generators and relations for C8×He3
G = < a,b,c,d | a8=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

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

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

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

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

C8×He3 is a maximal subgroup of
He33C16  He34C16  He34Q16  He35M4(2)  He36SD16  He34D8  He36M4(2)  He37SD16  He35D8  He35Q16

88 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 4A 4B 6A 6B 6C ··· 6J 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12T 24A ··· 24H 24I ··· 24AN order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 1 1 3 ··· 3 1 1 1 1 3 ··· 3 1 1 1 1 1 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C4 C6 C8 C12 C24 He3 C2×He3 C4×He3 C8×He3 kernel C8×He3 C4×He3 C3×C24 C2×He3 C3×C12 He3 C3×C6 C32 C8 C4 C2 C1 # reps 1 1 8 2 8 4 16 32 2 2 4 8

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

 63 0 0 0 63 0 0 0 63
,
 1 0 0 64 8 0 72 0 64
,
 8 0 0 0 8 0 0 0 8
,
 64 7 0 0 9 1 0 65 0
G:=sub<GL(3,GF(73))| [63,0,0,0,63,0,0,0,63],[1,64,72,0,8,0,0,0,64],[8,0,0,0,8,0,0,0,8],[64,0,0,7,9,65,0,1,0] >;

C8×He3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(216,19);
// by ID

G=gap.SmallGroup(216,19);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-2,108,386,165]);
// Polycyclic

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

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