p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C32, C16.27D4, C23.2C16, C2.2M6(2), C4.11M5(2), C8.25M4(2), (C2×C32)⋊1C2, (C2×C16).9C4, (C2×C4).3C16, (C2×C8).11C8, C2.1(C2×C32), (C22×C4).8C8, (C22×C8).25C4, C22.8(C2×C16), (C22×C16).6C2, C2.2(C22⋊C16), C4.33(C22⋊C8), C8.57(C22⋊C4), (C2×C16).109C22, (C2×C4).96(C2×C8), (C2×C8).260(C2×C4), SmallGroup(128,131)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22⋊C32
G = < a,b,c | a2=b2=c32=1, cac-1=ab=ba, bc=cb >
Smallest permutation representation of C22⋊C32
►On 64 points
Generators in S64
(1 17)(2 49)(3 19)(4 51)(5 21)(6 53)(7 23)(8 55)(9 25)(10 57)(11 27)(12 59)(13 29)(14 61)(15 31)(16 63)(18 33)(20 35)(22 37)(24 39)(26 41)(28 43)(30 45)(32 47)(34 50)(36 52)(38 54)(40 56)(42 58)(44 60)(46 62)(48 64)
(1 64)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)
(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)
G:=sub<Sym(64)| (1,17)(2,49)(3,19)(4,51)(5,21)(6,53)(7,23)(8,55)(9,25)(10,57)(11,27)(12,59)(13,29)(14,61)(15,31)(16,63)(18,33)(20,35)(22,37)(24,39)(26,41)(28,43)(30,45)(32,47)(34,50)(36,52)(38,54)(40,56)(42,58)(44,60)(46,62)(48,64), (1,64)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63), (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)>;
G:=Group( (1,17)(2,49)(3,19)(4,51)(5,21)(6,53)(7,23)(8,55)(9,25)(10,57)(11,27)(12,59)(13,29)(14,61)(15,31)(16,63)(18,33)(20,35)(22,37)(24,39)(26,41)(28,43)(30,45)(32,47)(34,50)(36,52)(38,54)(40,56)(42,58)(44,60)(46,62)(48,64), (1,64)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63), (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) );
G=PermutationGroup([[(1,17),(2,49),(3,19),(4,51),(5,21),(6,53),(7,23),(8,55),(9,25),(10,57),(11,27),(12,59),(13,29),(14,61),(15,31),(16,63),(18,33),(20,35),(22,37),(24,39),(26,41),(28,43),(30,45),(32,47),(34,50),(36,52),(38,54),(40,56),(42,58),(44,60),(46,62),(48,64)], [(1,64),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63)], [(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)]])
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 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | D4 | M4(2) | M5(2) | M6(2) |
| kernel | C22⋊C32 | C2×C32 | C22×C16 | C2×C16 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C16 | C8 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 2 | 2 | 4 | 8 |
Matrix representation of C22⋊C32 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 1 | 50 |
| 0 | 0 | 96 |
| 1 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 30 | 0 | 0 |
| 0 | 33 | 0 |
| 0 | 53 | 64 |
G:=sub<GL(3,GF(97))| [1,0,0,0,1,0,0,50,96],[1,0,0,0,96,0,0,0,96],[30,0,0,0,33,53,0,0,64] >;
C22⋊C32 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_{32} % in TeX
G:=Group("C2^2:C32"); // GroupNames label
G:=SmallGroup(128,131);
// by ID
G=gap.SmallGroup(128,131);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,80,102,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^32=1,c*a*c^-1=a*b=b*a,b*c=c*b>;
// generators/relations
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