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G = C2×M6(2)  order 128 = 27

Direct product of C2 and M6(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×M6(2)
 Chief series C1 — C2 — C4 — C8 — C16 — C2×C16 — C22×C16 — C2×M6(2)
 Lower central C1 — C2 — C2×M6(2)
 Upper central C1 — C2×C16 — C2×M6(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — C2×M6(2)

Generators and relations for C2×M6(2)
G = < a,b,c | a2=b32=c2=1, ab=ba, ac=ca, cbc=b17 >

Smallest permutation representation of C2×M6(2)
On 64 points
Generators in S64
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 33)
(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)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(33 49)(35 51)(37 53)(39 55)(41 57)(43 59)(45 61)(47 63)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 8I 8J 8K 8L 16A ··· 16P 16Q ··· 16X 32A ··· 32AF order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 8 8 8 8 16 ··· 16 16 ··· 16 32 ··· 32 size 1 1 1 1 2 2 1 1 1 1 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 C16 M6(2) kernel C2×M6(2) C2×C32 M6(2) C22×C16 C2×C16 C22×C8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 12 4 24 8 16

Matrix representation of C2×M6(2) in GL3(𝔽97) generated by

 96 0 0 0 1 0 0 0 1
,
 22 0 0 0 0 96 0 89 0
,
 96 0 0 0 1 0 0 0 96
G:=sub<GL(3,GF(97))| [96,0,0,0,1,0,0,0,1],[22,0,0,0,0,89,0,96,0],[96,0,0,0,1,0,0,0,96] >;

C2×M6(2) in GAP, Magma, Sage, TeX

C_2\times M_6(2)
% in TeX

G:=Group("C2xM6(2)");
// GroupNames label

G:=SmallGroup(128,989);
// by ID

G=gap.SmallGroup(128,989);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,56,925,80,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^17>;
// generators/relations

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