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G = C42.26D4  order 128 = 27

8th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.26D4, C4⋊C8.1C4, (C4×C8).7C4, C4.30C4≀C2, C22⋊C8.2C4, (C2×C4).35C42, C42.33(C2×C4), (C22×C4).20Q8, C23.13(C4⋊C4), (C22×C4).179D4, C2.7(C42⋊6C4), (C4×M4(2)).10C2, C42.6C4.8C2, (C2×C42).130C22, C2.2(C4.10C42), C2.7(M4(2)⋊4C4), C22.41(C2.C42), (C2×C4).18(C4⋊C4), (C22×C4).92(C2×C4), (C2×C4).297(C22⋊C4), SmallGroup(128,23)

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

Derived series
C1 — C2×C4 — C42.26D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — C42.26D4
Lower central
C1 — C2 — C2×C4 — C42.26D4
Upper central
C1 — C22 — C2×C42 — C42.26D4
Jennings
C1 — C22 — C22 — C2×C42 — C42.26D4

Generators and relations for C42.26D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=ab-1, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=ac3 >

Subgroups: 120 in 69 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×M4(2), C4×M4(2), C42.6C4, C42.26D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4≀C2, C4.10C42, C42⋊6C4, M4(2)⋊4C4, C42.26D4

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A···4J4K8A···8H8I···8P
order122224···448···88···8
size111142···244···48···8

32 irreducible representations

dim111111222244
type+++++-
imageC1C2C2C4C4C4D4D4Q8C4≀C2C4.10C42M4(2)⋊4C4
kernelC42.26D4C4×M4(2)C42.6C4C4×C8C22⋊C8C4⋊C8C42C22×C4C22×C4C4C2C2
# reps112444211822

Matrix representation of C42.26D4 ►in GL6(𝔽17)

400000
040000
0016900
0013100
0000169
0000131
,
1600000
0160000
0013000
0001300
000040
000004
,
400000
010000
000010
000001
004000
000400
,
010000
1300000
0000145
0000113
000500
006000

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,13,0,0,0,0,9,1,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,5,0,0,0,14,11,0,0,0,0,5,3,0,0] >;
 

C42.26D4 in GAP, Magma, Sage, TeX 

C_4^2._{26}D_4
 
% in TeX
 
G:=Group("C4^2.26D4");
 
// GroupNames label
 
G:=SmallGroup(128,23);
 
// by ID
 
G=gap.SmallGroup(128,23);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,248,3924,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a*b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*c^3>;
 
// generators/relations
 

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