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G = (C2xSD16):14C4order 128 = 27

10th semidirect product of C2xSD16 and C4 acting via C4/C2=C2

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

Aliases: C4.81(C4xD4), C4:C4.310D4, (C2xC4):12SD16, (C2xSD16):14C4, (C2xD4).204D4, C4.1(C4:D4), C2.10(C4xSD16), D4.2(C22:C4), C22.151(C4xD4), (C22xC4).687D4, C23.768(C2xD4), C22.4Q16:40C2, C2.2(D4.2D4), C2.5(C22:SD16), C22.93C22wrC2, C2.6(D4.7D4), C2.2(D4.D4), C22.55(C4oD8), (C22xC8).35C22, (C22xSD16).6C2, C22.58(C2xSD16), C22.76(C8:C22), (C2xC42).277C22, C2.10(SD16:C4), (C22xQ8).13C22, C22.114(C4:D4), (C22xC4).1364C23, C23.67C23:2C2, C22.7C42:27C2, C4.63(C22.D4), (C22xD4).460C22, C22.65(C8.C22), C2.16(C23.23D4), (C2xC8):20(C2xC4), (C2xQ8):6(C2xC4), (C2xC4xD4).21C2, (C2xQ8:C4):5C2, (C2xC4).996(C2xD4), C4.11(C2xC22:C4), (C2xD4:C4).3C2, (C2xD4).164(C2xC4), (C2xC4).561(C4oD4), (C2xC4:C4).765C22, (C2xC4).382(C22xC4), SmallGroup(128,609)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — (C2xSD16):14C4
C1C2C4C2xC4C22xC4C22xD4C2xC4xD4 — (C2xSD16):14C4
C1C2C2xC4 — (C2xSD16):14C4
C1C23C2xC42 — (C2xSD16):14C4
C1C2C2C22xC4 — (C2xSD16):14C4

Generators and relations for (C2xSD16):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, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C2.C42, D4:C4, Q8:C4, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C22xC8, C2xSD16, C2xSD16, C23xC4, C22xD4, C22xQ8, C22.7C42, C22.4Q16, C23.67C23, C2xD4:C4, C2xQ8:C4, C2xC4xD4, C22xSD16, (C2xSD16):14C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, SD16, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C2xSD16, C4oD8, C8:C22, C8.C22, C23.23D4, C4xSD16, SD16:C4, C22:SD16, D4.7D4, D4.D4, D4.2D4, (C2xSD16):14C4

Smallest permutation representation of (C2xSD16):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++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4SD16C4oD4C4oD8C8:C22C8.C22
kernel(C2xSD16):14C4C22.7C42C22.4Q16C23.67C23C2xD4:C4C2xQ8:C4C2xC4xD4C22xSD16C2xSD16C4:C4C22xC4C2xD4C2xC4C2xC4C22C22C22
# reps11111111822444411

Matrix representation of (C2xSD16):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] >;

(C2xSD16):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

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