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G = C42.106D4  order 128 = 27

88th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.106D4, C8.5(C4⋊C4), (C2×C8).11Q8, C8⋊C4.6C4, (C2×C8).109D4, C4.47(C4⋊Q8), C22.2(C4⋊Q8), C23.84(C2×Q8), (C22×C4).46Q8, C4.47(C4⋊1D4), C42.145(C2×C4), C2.10(C42⋊9C4), C4⋊M4(2).30C2, (C2×C42).260C22, (C22×C8).221C22, (C22×C4).1348C23, C2.10(M4(2).C4), (C2×M4(2)).170C22, C4.38(C2×C4⋊C4), (C2×C8).64(C2×C4), (C2×C8⋊C4).5C2, (C2×C4).49(C4⋊C4), (C2×C4).733(C2×D4), (C2×C4).198(C2×Q8), C22.107(C2×C4⋊C4), (C2×C8.C4).12C2, (C2×C4).547(C22×C4), SmallGroup(128,581)

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

Derived series
C1 — C2×C4 — C42.106D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.106D4
Lower central
C1 — C2 — C2×C4 — C42.106D4
Upper central
C1 — C2×C4 — C2×C42 — C42.106D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.106D4

Generators and relations for C42.106D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 156 in 106 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C8⋊C4, C4⋊C8, C8.C4, C2×C42, C22×C8, C2×M4(2), C2×C8⋊C4, C4⋊M4(2), C2×C8.C4, C42.106D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C42⋊9C4, M4(2).C4, C42.106D4

Smallest permutation representation of C42.106D4
►On 64 points
Generators in S64
(1 40 27 16)(2 37 28 13)(3 34 29 10)(4 39 30 15)(5 36 31 12)(6 33 32 9)(7 38 25 14)(8 35 26 11)(17 43 55 58)(18 48 56 63)(19 45 49 60)(20 42 50 57)(21 47 51 62)(22 44 52 59)(23 41 53 64)(24 46 54 61)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I···8P
order12222244444444448···88···8
size11112211112244444···48···8

32 irreducible representations

dim1111122224
type++++++--
imageC1C2C2C2C4D4D4Q8Q8M4(2).C4
kernelC42.106D4C2×C8⋊C4C4⋊M4(2)C2×C8.C4C8⋊C4C42C2×C8C2×C8C22×C4C2
# reps1124824424

Matrix representation of C42.106D4 ►in GL6(𝔽17)

0160000
100000
000100
0016000
0000016
000010
,
1600000
0160000
0013000
0001300
0000130
0000013
,
1600000
0160000
000200
002000
000008
000080
,
1670000
710000
000008
000080
000900
009000

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[16,7,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,8,0,0,0,0,8,0,0,0] >;
 

C42.106D4 in GAP, Magma, Sage, TeX 

C_4^2._{106}D_4
 
% in TeX
 
G:=Group("C4^2.106D4");
 
// GroupNames label
 
G:=SmallGroup(128,581);
 
// by ID
 
G=gap.SmallGroup(128,581);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,2019,248,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

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