p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22.3M5(2), (C2×C4)⋊C16, (C2×C8).292D4, (C2×C42).7C4, (C22×C8).7C4, (C22×C4).3C8, C22⋊C16.1C2, C23.27(C2×C8), C22.3(C2×C16), C4.39(C23⋊C4), C2.4(C22⋊C16), C2.2(C23.C8), (C2×C4).55M4(2), (C22×C8).3C22, C4.20(C4.10D4), C22.34(C22⋊C8), C2.2(C22.M4(2)), (C2×C4⋊C8).8C2, (C22×C4).428(C2×C4), (C2×C4).378(C22⋊C4), SmallGroup(128,54)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.M5(2)
G = < a,b,c,d | a2=b2=c16=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc9 >
(1 9)(2 48)(3 11)(4 34)(5 13)(6 36)(7 15)(8 38)(10 40)(12 42)(14 44)(16 46)(17 25)(18 62)(19 27)(20 64)(21 29)(22 50)(23 31)(24 52)(26 54)(28 56)(30 58)(32 60)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 49)(30 50)(31 51)(32 52)
(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)
(1 55 39 19)(2 20 40 56)(3 21 41 57)(4 58 42 22)(5 59 43 23)(6 24 44 60)(7 25 45 61)(8 62 46 26)(9 63 47 27)(10 28 48 64)(11 29 33 49)(12 50 34 30)(13 51 35 31)(14 32 36 52)(15 17 37 53)(16 54 38 18)
G:=sub<Sym(64)| (1,9)(2,48)(3,11)(4,34)(5,13)(6,36)(7,15)(8,38)(10,40)(12,42)(14,44)(16,46)(17,25)(18,62)(19,27)(20,64)(21,29)(22,50)(23,31)(24,52)(26,54)(28,56)(30,58)(32,60)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,49)(30,50)(31,51)(32,52), (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), (1,55,39,19)(2,20,40,56)(3,21,41,57)(4,58,42,22)(5,59,43,23)(6,24,44,60)(7,25,45,61)(8,62,46,26)(9,63,47,27)(10,28,48,64)(11,29,33,49)(12,50,34,30)(13,51,35,31)(14,32,36,52)(15,17,37,53)(16,54,38,18)>;
G:=Group( (1,9)(2,48)(3,11)(4,34)(5,13)(6,36)(7,15)(8,38)(10,40)(12,42)(14,44)(16,46)(17,25)(18,62)(19,27)(20,64)(21,29)(22,50)(23,31)(24,52)(26,54)(28,56)(30,58)(32,60)(33,41)(35,43)(37,45)(39,47)(49,57)(51,59)(53,61)(55,63), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,49)(30,50)(31,51)(32,52), (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), (1,55,39,19)(2,20,40,56)(3,21,41,57)(4,58,42,22)(5,59,43,23)(6,24,44,60)(7,25,45,61)(8,62,46,26)(9,63,47,27)(10,28,48,64)(11,29,33,49)(12,50,34,30)(13,51,35,31)(14,32,36,52)(15,17,37,53)(16,54,38,18) );
G=PermutationGroup([[(1,9),(2,48),(3,11),(4,34),(5,13),(6,36),(7,15),(8,38),(10,40),(12,42),(14,44),(16,46),(17,25),(18,62),(19,27),(20,64),(21,29),(22,50),(23,31),(24,52),(26,54),(28,56),(30,58),(32,60),(33,41),(35,43),(37,45),(39,47),(49,57),(51,59),(53,61),(55,63)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,49),(30,50),(31,51),(32,52)], [(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)], [(1,55,39,19),(2,20,40,56),(3,21,41,57),(4,58,42,22),(5,59,43,23),(6,24,44,60),(7,25,45,61),(8,62,46,26),(9,63,47,27),(10,28,48,64),(11,29,33,49),(12,50,34,30),(13,51,35,31),(14,32,36,52),(15,17,37,53),(16,54,38,18)]])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | - | |||||||
image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | M4(2) | M5(2) | C23⋊C4 | C4.10D4 | C23.C8 |
kernel | C22.M5(2) | C22⋊C16 | C2×C4⋊C8 | C2×C42 | C22×C8 | C22×C4 | C2×C4 | C2×C8 | C2×C4 | C22 | C4 | C4 | C2 |
# reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C22.M5(2) ►in GL6(𝔽17)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 3 | 12 | 16 | 0 |
0 | 0 | 3 | 15 | 0 | 16 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
7 | 7 | 0 | 0 | 0 | 0 |
3 | 10 | 0 | 0 | 0 | 0 |
0 | 0 | 11 | 10 | 4 | 0 |
0 | 0 | 7 | 6 | 9 | 9 |
0 | 0 | 16 | 4 | 9 | 3 |
0 | 0 | 16 | 5 | 14 | 8 |
16 | 0 | 0 | 0 | 0 | 0 |
2 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 13 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 14 | 13 | 0 |
0 | 0 | 10 | 3 | 8 | 4 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,3,3,0,0,0,1,12,15,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[7,3,0,0,0,0,7,10,0,0,0,0,0,0,11,7,16,16,0,0,10,6,4,5,0,0,4,9,9,14,0,0,0,9,3,8],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,10,0,0,0,4,14,3,0,0,0,0,13,8,0,0,0,0,0,4] >;
C22.M5(2) in GAP, Magma, Sage, TeX
C_2^2.M_5(2)
% in TeX
G:=Group("C2^2.M5(2)");
// GroupNames label
G:=SmallGroup(128,54);
// by ID
G=gap.SmallGroup(128,54);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,346,136,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b*c^9>;
// generators/relations
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