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G = C42.418D4order 128 = 27

51st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.418D4, C42.172C23, (C4×Q8).8C4, C22⋊Q8.15C4, C4.102(C4○D8), C4.10D838C2, C4⋊C8.262C22, C42.113(C2×C4), C4.6Q1623C2, (C22×C4).241D4, C4⋊Q8.245C22, C4.106(C8.C22), C23.65(C22⋊C4), C42.6C4.24C2, (C2×C42).216C22, C42.12C4.27C2, C2.13(C23.38D4), C2.19(C23.24D4), C23.37C23.16C2, C2.19(M4(2).8C22), C4⋊C4.42(C2×C4), (C2×Q8).33(C2×C4), (C2×C4).1243(C2×D4), (C2×C4).166(C22×C4), (C22×C4).238(C2×C4), (C2×C4).187(C22⋊C4), C22.230(C2×C22⋊C4), SmallGroup(128,286)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.418D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.418D4
C1C22C2×C4 — C42.418D4
C1C22C2×C42 — C42.418D4
C1C22C22C42 — C42.418D4

Generators and relations for C42.418D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 180 in 99 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×8], C22, C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C4.10D8 [×2], C4.6Q16 [×2], C42.12C4, C42.6C4, C23.37C23, C42.418D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C4○D8 [×2], C8.C22 [×2], M4(2).8C22, C23.24D4, C23.38D4, C42.418D4

Smallest permutation representation of C42.418D4
On 64 points
Generators in S64
(1 3 5 7)(2 60 6 64)(4 62 8 58)(9 11 13 15)(10 20 14 24)(12 22 16 18)(17 19 21 23)(25 31 29 27)(26 45 30 41)(28 47 32 43)(33 52 37 56)(34 40 38 36)(35 54 39 50)(42 48 46 44)(49 55 53 51)(57 59 61 63)
(1 15 61 19)(2 20 62 16)(3 9 63 21)(4 22 64 10)(5 11 57 23)(6 24 58 12)(7 13 59 17)(8 18 60 14)(25 51 42 34)(26 35 43 52)(27 53 44 36)(28 37 45 54)(29 55 46 38)(30 39 47 56)(31 49 48 40)(32 33 41 50)
(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 48 15 40 61 31 19 49)(2 52 20 26 62 35 16 43)(3 46 9 38 63 29 21 55)(4 50 22 32 64 33 10 41)(5 44 11 36 57 27 23 53)(6 56 24 30 58 39 12 47)(7 42 13 34 59 25 17 51)(8 54 18 28 60 37 14 45)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A···4J4K4L4M4N4O8A···8H8I8J8K8L
order122224···4444448···88888
size111142···2488884···48888

32 irreducible representations

dim1111111122244
type++++++++-
imageC1C2C2C2C2C2C4C4D4D4C4○D8C8.C22M4(2).8C22
kernelC42.418D4C4.10D8C4.6Q16C42.12C4C42.6C4C23.37C23C4×Q8C22⋊Q8C42C22×C4C4C4C2
# reps1221114422822

Matrix representation of C42.418D4 in GL6(𝔽17)

100000
0160000
0013000
0001300
0000130
0000013
,
1300000
040000
004004
0001300
000044
0000013
,
0150000
800000
0013006
00401610
00131300
008004
,
800000
0150000
00013111
000406
0041304
0008013

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,4,0,4,13],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,13,4,13,8,0,0,0,0,13,0,0,0,0,16,0,0,0,0,6,10,0,4],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,4,0,0,0,13,4,13,8,0,0,1,0,0,0,0,0,11,6,4,13] >;

C42.418D4 in GAP, Magma, Sage, TeX

C_4^2._{418}D_4
% in TeX

G:=Group("C4^2.418D4");
// GroupNames label

G:=SmallGroup(128,286);
// by ID

G=gap.SmallGroup(128,286);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,520,1123,1018,248,1971,242]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations

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