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G = C42.258D4  order 128 = 27

240th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.258D4, C42.393C23, C8⋊Q8⋊11C2, C8⋊8D4.4C2, C8.2D4⋊8C2, C8.31(C4○D4), Q16⋊C4⋊16C2, C8.18D4⋊28C2, C4⋊C4.120C23, (C2×C8).281C23, (C2×C4).379C24, (C4×D4).99C22, C23.398(C2×D4), (C22×C4).478D4, C4⋊Q8.295C22, SD16⋊C4⋊24C2, (C4×Q8).96C22, C4.Q8.31C22, C2.D8.98C22, (C2×D4).133C23, (C2×Q8).121C23, (C2×Q16).66C22, C8⋊C4.136C22, C4⋊D4.176C22, (C2×C42).865C22, (C22×C8).281C22, (C2×SD16).27C22, C22.639(C22×D4), C22.1(C8.C22), C22⋊Q8.181C22, D4⋊C4.137C22, C2.45(D8⋊C22), (C22×C4).1575C23, Q8⋊C4.130C22, C4.4D4.147C22, C42.C2.124C22, C42.28C22⋊36C2, C42.30C22⋊22C2, C23.37C23⋊14C2, C23.36C23.23C2, C2.76(C22.26C24), (C2×C8⋊C4)⋊12C2, C4.64(C2×C4○D4), (C2×C4).703(C2×D4), C2.46(C2×C8.C22), SmallGroup(128,1913)

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

Derived series
C1 — C2×C4 — C42.258D4
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C8⋊C4 — C2×C8⋊C4 — C42.258D4
Lower central
C1 — C2 — C2×C4 — C42.258D4
Upper central
C1 — C22 — C2×C42 — C42.258D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.258D4

Generators and relations for C42.258D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=ab2, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 316 in 183 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C8⋊C4, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×SD16, C2×Q16, C2×C8⋊C4, SD16⋊C4, Q16⋊C4, C8⋊8D4, C8.18D4, C42.28C22, C42.30C22, C8.2D4, C8⋊Q8, C23.36C23, C23.37C23, C42.258D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, C22.26C24, C2×C8.C22, D8⋊C22, C42.258D4

Smallest permutation representation of C42.258D4
►On 64 points
Generators in S64
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I4J4K···4Q8A···8H
order12222224···4444···48···8
size11112282···2448···84···4

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C8.C22D8⋊C22
kernelC42.258D4C2×C8⋊C4SD16⋊C4Q16⋊C4C8⋊8D4C8.18D4C42.28C22C42.30C22C8.2D4C8⋊Q8C23.36C23C23.37C23C42C22×C4C8C22C2
# reps11222211111122822

Matrix representation of C42.258D4 ►in GL6(𝔽17)

480000
13130000
0011500
0011600
0000115
0000116
,
400000
040000
0013000
0001300
000040
000004
,
16150000
110000
000010
000001
0011500
0011600
,
100000
16160000
0016200
000100
000010
0000116

G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,8,13,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
 

C42.258D4 in GAP, Magma, Sage, TeX 

C_4^2._{258}D_4
 
% in TeX
 
G:=Group("C4^2.258D4");
 
// GroupNames label
 
G:=SmallGroup(128,1913);
 
// by ID
 
G=gap.SmallGroup(128,1913);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,184,521,80,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
 
// generators/relations
 

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