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G = C42.26D4order 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(C426C4), (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

C1C2×C4 — C42.26D4
C1C2C22C2×C4C22×C4C2×C42C4×M4(2) — C42.26D4
C1C2C2×C4 — C42.26D4
C1C22C2×C42 — C42.26D4
C1C22C22C2×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 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×8], C2×C4 [×6], C2×C4 [×4], C23, C42 [×4], C2×C8 [×6], M4(2) [×4], C22×C4 [×3], C4×C8 [×2], C8⋊C4 [×3], C22⋊C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C2×M4(2), C4×M4(2), C42.6C4 [×2], C42.26D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×2], C4.10C42, C426C4, M4(2)⋊4C4, C42.26D4

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

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

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

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

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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