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G = (C2×C4)⋊5SD16  order 128 = 27

3rd semidirect product of C2×C4 and SD16 acting via SD16/C4=C22

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

Aliases: (C2×C4)⋊5SD16, (C2×C8).159D4, C4.77C22≀C2, (C2×D4).116D4, (C2×Q8).106D4, C2.19(C8⋊D4), C2.19(C8⋊8D4), C23.917(C2×D4), (C22×C4).151D4, C22.4Q16⋊43C2, C2.14(C4⋊SD16), C4.73(C4.4D4), (C22×C8).78C22, C22.99(C2×SD16), C2.14(D4.D4), C2.19(D4.2D4), C22.113(C4○D8), (C2×C42).369C22, C2.18(Q8.D4), (C22×SD16).12C2, (C22×D4).87C22, (C22×Q8).72C22, C22.239(C4⋊D4), C22.141(C8⋊C22), (C22×C4).1451C23, C4.77(C22.D4), C23.67C23⋊10C2, C2.28(C23.10D4), C22.129(C8.C22), C24.3C22.16C2, (C2×C4⋊C8)⋊33C2, (C2×Q8⋊C4)⋊16C2, (C2×C4).1046(C2×D4), (C2×D4⋊C4).14C2, (C2×C4).882(C4○D4), (C2×C4⋊C4).132C22, SmallGroup(128,787)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22×C4 — (C2×C4)⋊5SD16
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×D4⋊C4 — (C2×C4)⋊5SD16
Lower central
C1 — C2 — C22×C4 — (C2×C4)⋊5SD16
Upper central
C1 — C23 — C2×C42 — (C2×C4)⋊5SD16
Jennings
C1 — C2 — C2 — C22×C4 — (C2×C4)⋊5SD16

Generators and relations for (C2×C4)⋊5SD16
 G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd=c3 >

Subgroups: 384 in 162 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×SD16, C22×D4, C22×Q8, C22.4Q16, C24.3C22, C23.67C23, C2×D4⋊C4, C2×Q8⋊C4, C2×C4⋊C8, C22×SD16, (C2×C4)⋊5SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.10D4, C4⋊SD16, D4.D4, D4.2D4, Q8.D4, C8⋊8D4, C8⋊D4, (C2×C4)⋊5SD16

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

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

(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)

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim11111111222222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D4SD16C4○D4C4○D8C8⋊C22C8.C22
kernel(C2×C4)⋊5SD16C22.4Q16C24.3C22C23.67C23C2×D4⋊C4C2×Q8⋊C4C2×C4⋊C8C22×SD16C2×C8C22×C4C2×D4C2×Q8C2×C4C2×C4C22C22C22
# reps11111111222246411

Matrix representation of (C2×C4)⋊5SD16 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
13150000
040000
0016000
0001600
000001
000010
,
100000
13160000
0012500
00121200
0000160
0000016
,
100000
13160000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,15,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

(C2×C4)⋊5SD16 in GAP, Magma, Sage, TeX 

(C_2\times C_4)\rtimes_5{\rm SD}_{16}
 
% in TeX
 
G:=Group("(C2xC4):5SD16");
 
// GroupNames label
 
G:=SmallGroup(128,787);
 
// by ID
 
G=gap.SmallGroup(128,787);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2019,1018,248,2804,1411,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d=c^3>;
 
// generators/relations
 

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