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G = C22⋊C32order 128 = 27

The semidirect product of C22 and C32 acting via C32/C16=C2

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

C1C2 — C22⋊C32
C1C2C4C8C16C2×C16C22×C16 — C22⋊C32
C1C2 — C22⋊C32
C1C2×C16 — C22⋊C32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×C16 — C22⋊C32

Generators and relations for C22⋊C32
 G = < a,b,c | a2=b2=c32=1, cac-1=ab=ba, bc=cb >

2C2
2C2
2C22
2C4
2C22
2C8
2C2×C4
2C2×C4
2C2×C8
2C2×C8
2C16
2C32
2C2×C16
2C2×C16
2C32

Smallest permutation representation of C22⋊C32
On 64 points
Generators in S64
(1 17)(2 51)(3 19)(4 53)(5 21)(6 55)(7 23)(8 57)(9 25)(10 59)(11 27)(12 61)(13 29)(14 63)(15 31)(16 33)(18 35)(20 37)(22 39)(24 41)(26 43)(28 45)(30 47)(32 49)(34 50)(36 52)(38 54)(40 56)(42 58)(44 60)(46 62)(48 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)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H8I8J8K8L16A···16P16Q···16X32A···32AF
order1222224444448···8888816···1616···1632···32
size1111221111221···122221···12···22···2

80 irreducible representations

dim11111111112222
type++++
imageC1C2C2C4C4C8C8C16C16C32D4M4(2)M5(2)M6(2)
kernelC22⋊C32C2×C32C22×C16C2×C16C22×C8C2×C8C22×C4C2×C4C23C22C16C8C4C2
# reps121224488322248

Matrix representation of C22⋊C32 in GL3(𝔽97) generated by

100
0150
0096
,
100
0960
0096
,
3000
0330
05364
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

Export

Subgroup lattice of C22⋊C32 in TeX

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