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G = C42.81D4order 128 = 27

63rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.81D4, C42.170C23, C4.100(C4○D8), C4.10D836C2, C4⋊C8.207C22, C4.85(C8⋊C22), C42⋊C2.7C4, C42.111(C2×C4), (C22×C4).746D4, C4⋊Q8.243C22, C42.C2.13C4, C4.87(C8.C22), C42.6C4.23C2, (C2×C42).214C22, C23.112(C22⋊C4), C22.3(C4.10D4), C2.17(C23.24D4), C2.15(C23.36D4), C23.37C23.15C2, (C2×C4⋊C8).15C2, C4⋊C4.40(C2×C4), (C2×C4).1241(C2×D4), (C22×C4).236(C2×C4), (C2×C4).164(C22×C4), C2.19(C2×C4.10D4), (C2×C4).185(C22⋊C4), C22.228(C2×C22⋊C4), SmallGroup(128,284)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.81D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.81D4
C1C22C2×C4 — C42.81D4
C1C22C2×C42 — C42.81D4
C1C22C22C42 — C42.81D4

Generators and relations for C42.81D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 188 in 104 conjugacy classes, 48 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×10], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×6], C22×C4 [×3], C2×Q8 [×2], C8⋊C4, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C22×C8, C4.10D8 [×4], C2×C4⋊C8, C42.6C4, C23.37C23, C42.81D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C4○D8 [×2], C8⋊C22, C8.C22, C2×C4.10D4, C23.24D4, C23.36D4, C42.81D4

Smallest permutation representation of C42.81D4
On 64 points
Generators in S64
(1 35 54 63)(2 40 55 60)(3 37 56 57)(4 34 49 62)(5 39 50 59)(6 36 51 64)(7 33 52 61)(8 38 53 58)(9 23 27 43)(10 20 28 48)(11 17 29 45)(12 22 30 42)(13 19 31 47)(14 24 32 44)(15 21 25 41)(16 18 26 46)
(1 61 50 37)(2 38 51 62)(3 63 52 39)(4 40 53 64)(5 57 54 33)(6 34 55 58)(7 59 56 35)(8 36 49 60)(9 21 31 45)(10 46 32 22)(11 23 25 47)(12 48 26 24)(13 17 27 41)(14 42 28 18)(15 19 29 43)(16 44 30 20)
(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 61 26 50 24 37 12)(2 15 38 19 51 29 62 43)(3 46 63 32 52 22 39 10)(4 13 40 17 53 27 64 41)(5 44 57 30 54 20 33 16)(6 11 34 23 55 25 58 47)(7 42 59 28 56 18 35 14)(8 9 36 21 49 31 60 45)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N8A···8H8I8J8K8L
order1222224···44444448···88888
size1111222···24488884···48888

32 irreducible representations

dim1111111222444
type++++++++--
imageC1C2C2C2C2C4C4D4D4C4○D8C8⋊C22C8.C22C4.10D4
kernelC42.81D4C4.10D8C2×C4⋊C8C42.6C4C23.37C23C42⋊C2C42.C2C42C22×C4C4C4C4C22
# reps1411144228112

Matrix representation of C42.81D4 in GL6(𝔽17)

400000
040000
0013020
00161215
001040
00110416
,
1300000
240000
0016200
0016100
00165115
0064116
,
16130000
010000
001551415
0055112
0071610
00510513
,
900000
1120000
0011716
001091510
00151007
00151057

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,16,1,11,0,0,0,1,0,0,0,0,2,2,4,4,0,0,0,15,0,16],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,16,16,16,6,0,0,2,1,5,4,0,0,0,0,1,1,0,0,0,0,15,16],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,15,5,7,5,0,0,5,5,16,10,0,0,14,1,1,5,0,0,15,12,0,13],[9,11,0,0,0,0,0,2,0,0,0,0,0,0,1,10,15,15,0,0,1,9,10,10,0,0,7,15,0,5,0,0,16,10,7,7] >;

C42.81D4 in GAP, Magma, Sage, TeX

C_4^2._{81}D_4
% in TeX

G:=Group("C4^2.81D4");
// GroupNames label

G:=SmallGroup(128,284);
// by ID

G=gap.SmallGroup(128,284);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,758,352,1123,1018,248,1971,242]);
// Polycyclic

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

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