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G = C42.27D4  order 128 = 27

9th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.27D4, C4⋊C8.2C4, (C2×C4).9D8, (C2×C4).5Q16, (C22×C8).3C4, (C2×C4).36C42, C42.34(C2×C4), (C2×C4).12SD16, (C22×C4).21Q8, C23.40(C4⋊C4), (C22×C4).180D4, C4.29(D4⋊C4), C22.8(C4.Q8), C22.9(C2.D8), C4.21(Q8⋊C4), C4.10(C4.D4), C4⋊M4(2).5C2, C4.10(C4.10D4), C2.5(C22.4Q16), (C2×C42).131C22, C2.3(C4.10C42), C2.5(C22.C42), C22.42(C2.C42), (C2×C4⋊C8).5C2, (C2×C4).19(C4⋊C4), (C22×C4).465(C2×C4), (C2×C4).338(C22⋊C4), SmallGroup(128,24)

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

Derived series
C1 — C2×C4 — C42.27D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊C8 — C42.27D4
Lower central
C1 — C2 — C2×C4 — C42.27D4
Upper central
C1 — C22 — C2×C42 — C42.27D4
Jennings
C1 — C22 — C22 — C2×C42 — C42.27D4

Generators and relations for C42.27D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >

Subgroups: 136 in 76 conjugacy classes, 42 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, M4(2), C22×C4, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C2×M4(2), C2×C4⋊C8, C4⋊M4(2), C42.27D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, C4.D4, C4.10D4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4.10C42, C22.4Q16, C22.C42, C42.27D4

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J8A···8H8I···8P
order1222224···4448···88···8
size1111222···2444···48···8

32 irreducible representations

dim11111222222444
type+++++-+-+-
imageC1C2C2C4C4D4D4Q8D8SD16Q16C4.D4C4.10D4C4.10C42
kernelC42.27D4C2×C4⋊C8C4⋊M4(2)C4⋊C8C22×C8C42C22×C4C22×C4C2×C4C2×C4C2×C4C4C4C2
# reps11284211242112

Matrix representation of C42.27D4 ►in GL6(𝔽17)

100000
010000
000100
0016000
000001
0000160
,
1610000
1510000
0013000
0001300
000040
000004
,
8160000
1490000
000009
000090
0015000
000200
,
2110000
9150000
000010
000001
000100
0016000

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,15,0,0,0,0,1,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[8,14,0,0,0,0,16,9,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,2,0,0,0,9,0,0,0,0,9,0,0,0],[2,9,0,0,0,0,11,15,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42.27D4 in GAP, Magma, Sage, TeX 

C_4^2._{27}D_4
 
% in TeX
 
G:=Group("C4^2.27D4");
 
// GroupNames label
 
G:=SmallGroup(128,24);
 
// by ID
 
G=gap.SmallGroup(128,24);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,248,3924,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b*c^3>;
 
// generators/relations
 

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