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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

Aliases: C2×M6(2), C4M6(2), C8M6(2), C324C22, C16M6(2), C23.3C16, C16.16C23, (C2×C32)⋊8C2, C8.26(C2×C8), (C2×C8).19C8, (C2×C4).6C16, C16.23(C2×C4), (C2×C16).19C4, C4.10(C2×C16), (C22×C8).49C4, C8.68(C22×C4), (C22×C4).17C8, C2.6(C22×C16), C4.37(C22×C8), (C22×C16).18C2, C22.11(C2×C16), (C2×C16).106C22, (C2×C4).101(C2×C8), (C2×C8).254(C2×C4), SmallGroup(128,989)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×M6(2)
C1C2C4C8C16C2×C16C22×C16 — C2×M6(2)
C1C2 — C2×M6(2)
C1C2×C16 — C2×M6(2)
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C2×M6(2)

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

2C2
2C2
2C22
2C22

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 2A2B2C2D2E4A4B4C4D4E4F8A···8H8I8J8K8L16A···16P16Q···16X32A···32AF
order1222224444448···8888816···1616···1632···32
size1111221111221···122221···12···22···2

80 irreducible representations

dim11111111112
type++++
imageC1C2C2C2C4C4C8C8C16C16M6(2)
kernelC2×M6(2)C2×C32M6(2)C22×C16C2×C16C22×C8C2×C8C22×C4C2×C4C23C2
# reps12416212424816

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

9600
010
001
,
2200
0096
0890
,
9600
010
0096
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

Export

Subgroup lattice of C2×M6(2) in TeX

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