Copied to
clipboard

G = (C2×SD16)⋊14C4order 128 = 27

10th semidirect product of C2×SD16 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C4.81(C4×D4), C4⋊C4.310D4, (C2×C4)⋊12SD16, (C2×SD16)⋊14C4, (C2×D4).204D4, C4.1(C4⋊D4), C2.10(C4×SD16), D4.2(C22⋊C4), C22.151(C4×D4), (C22×C4).687D4, C23.768(C2×D4), C22.4Q1640C2, C2.2(D4.2D4), C2.5(C22⋊SD16), C22.93C22≀C2, C2.6(D4.7D4), C2.2(D4.D4), C22.55(C4○D8), (C22×C8).35C22, (C22×SD16).6C2, C22.58(C2×SD16), C22.76(C8⋊C22), (C2×C42).277C22, C2.10(SD16⋊C4), (C22×Q8).13C22, C22.114(C4⋊D4), (C22×C4).1364C23, C23.67C232C2, C22.7C4227C2, C4.63(C22.D4), (C22×D4).460C22, C22.65(C8.C22), C2.16(C23.23D4), (C2×C8)⋊20(C2×C4), (C2×Q8)⋊6(C2×C4), (C2×C4×D4).21C2, (C2×Q8⋊C4)⋊5C2, (C2×C4).996(C2×D4), C4.11(C2×C22⋊C4), (C2×D4⋊C4).3C2, (C2×D4).164(C2×C4), (C2×C4).561(C4○D4), (C2×C4⋊C4).765C22, (C2×C4).382(C22×C4), SmallGroup(128,609)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×SD16)⋊14C4
C1C2C4C2×C4C22×C4C22×D4C2×C4×D4 — (C2×SD16)⋊14C4
C1C2C2×C4 — (C2×SD16)⋊14C4
C1C23C2×C42 — (C2×SD16)⋊14C4
C1C2C2C22×C4 — (C2×SD16)⋊14C4

Generators and relations for (C2×SD16)⋊14C4
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=ab-1, dcd-1=b4c >

Subgroups: 436 in 201 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×8], C22 [×7], C22 [×16], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×24], D4 [×4], D4 [×6], Q8 [×6], C23, C23 [×10], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×5], SD16 [×8], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C22×C8 [×2], C2×SD16 [×4], C2×SD16 [×4], C23×C4, C22×D4, C22×Q8, C22.7C42, C22.4Q16, C23.67C23, C2×D4⋊C4, C2×Q8⋊C4, C2×C4×D4, C22×SD16, (C2×SD16)⋊14C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.23D4, C4×SD16, SD16⋊C4, C22⋊SD16, D4.7D4, D4.D4, D4.2D4, (C2×SD16)⋊14C4

Smallest permutation representation of (C2×SD16)⋊14C4
On 64 points
Generators in S64
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)
(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 49)(2 52)(3 55)(4 50)(5 53)(6 56)(7 51)(8 54)(9 45)(10 48)(11 43)(12 46)(13 41)(14 44)(15 47)(16 42)(17 38)(18 33)(19 36)(20 39)(21 34)(22 37)(23 40)(24 35)(25 60)(26 63)(27 58)(28 61)(29 64)(30 59)(31 62)(32 57)
(1 43 59 38)(2 10 60 24)(3 41 61 36)(4 16 62 22)(5 47 63 34)(6 14 64 20)(7 45 57 40)(8 12 58 18)(9 28 23 55)(11 26 17 53)(13 32 19 51)(15 30 21 49)(25 39 52 44)(27 37 54 42)(29 35 56 48)(31 33 50 46)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4N4O4P4Q4R8A···8H
order12···222224···44···444448···8
size11···144442···24···488884···4

38 irreducible representations

dim11111111122222244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4SD16C4○D4C4○D8C8⋊C22C8.C22
kernel(C2×SD16)⋊14C4C22.7C42C22.4Q16C23.67C23C2×D4⋊C4C2×Q8⋊C4C2×C4×D4C22×SD16C2×SD16C4⋊C4C22×C4C2×D4C2×C4C2×C4C22C22C22
# reps11111111822444411

Matrix representation of (C2×SD16)⋊14C4 in GL5(𝔽17)

10000
016000
001600
000160
000016
,
10000
0121200
051200
000016
00010
,
160000
01000
001600
000160
00001
,
40000
001300
013000
000160
000016

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,12,5,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,16,0],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,16] >;

(C2×SD16)⋊14C4 in GAP, Magma, Sage, TeX

(C_2\times {\rm SD}_{16})\rtimes_{14}C_4
% in TeX

G:=Group("(C2xSD16):14C4");
// GroupNames label

G:=SmallGroup(128,609);
// by ID

G=gap.SmallGroup(128,609);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=a*b^-1,d*c*d^-1=b^4*c>;
// generators/relations

׿
×
𝔽