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G = C42.80D4order 128 = 27

62nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.80D4, C42.169C23, C4.99(C4○D8), C4.D822C2, C4⋊C8.206C22, C4.84(C8⋊C22), C42.110(C2×C4), C4.6Q1622C2, C4.4D4.11C4, (C22×C4).745D4, C4⋊Q8.242C22, C42.6C440C2, C4.86(C8.C22), C41D4.128C22, (C2×C42).213C22, C22.3(C4.D4), C23.111(C22⋊C4), C2.14(C23.36D4), C2.16(C23.24D4), C22.26C24.15C2, (C2×C4⋊C8)⋊8C2, (C2×C4○D4).7C4, (C2×D4).32(C2×C4), (C2×Q8).32(C2×C4), (C2×C4).1240(C2×D4), C2.20(C2×C4.D4), (C22×C4).235(C2×C4), (C2×C4).163(C22×C4), (C2×C4).184(C22⋊C4), C22.227(C2×C22⋊C4), SmallGroup(128,283)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.80D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.80D4
C1C22C2×C4 — C42.80D4
C1C22C2×C42 — C42.80D4
C1C22C22C42 — C42.80D4

Generators and relations for C42.80D4
 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=a2b-1c3 >

Subgroups: 284 in 126 conjugacy classes, 48 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×12], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×6], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C8⋊C4, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8, C2×C4○D4 [×2], C4.D8 [×2], C4.6Q16 [×2], C2×C4⋊C8, C42.6C4, C22.26C24, C42.80D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C4○D8 [×2], C8⋊C22, C8.C22, C2×C4.D4, C23.24D4, C23.36D4, C42.80D4

Smallest permutation representation of C42.80D4
On 64 points
Generators in S64
(1 31 15 41)(2 28 16 46)(3 25 9 43)(4 30 10 48)(5 27 11 45)(6 32 12 42)(7 29 13 47)(8 26 14 44)(17 50 40 61)(18 55 33 58)(19 52 34 63)(20 49 35 60)(21 54 36 57)(22 51 37 62)(23 56 38 59)(24 53 39 64)
(1 50 11 57)(2 58 12 51)(3 52 13 59)(4 60 14 53)(5 54 15 61)(6 62 16 55)(7 56 9 63)(8 64 10 49)(17 27 36 41)(18 42 37 28)(19 29 38 43)(20 44 39 30)(21 31 40 45)(22 46 33 32)(23 25 34 47)(24 48 35 26)
(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 64 50 10 11 49 57 8)(2 7 58 56 12 9 51 63)(3 62 52 16 13 55 59 6)(4 5 60 54 14 15 53 61)(17 26 27 24 36 48 41 35)(18 34 42 47 37 23 28 25)(19 32 29 22 38 46 43 33)(20 40 44 45 39 21 30 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L8A···8H8I8J8K8L
order122222224···444448···88888
size111122882···244884···48888

32 irreducible representations

dim11111111222444
type+++++++++-+
imageC1C2C2C2C2C2C4C4D4D4C4○D8C8⋊C22C8.C22C4.D4
kernelC42.80D4C4.D8C4.6Q16C2×C4⋊C8C42.6C4C22.26C24C4.4D4C2×C4○D4C42C22×C4C4C4C4C22
# reps12211144228112

Matrix representation of C42.80D4 in GL6(𝔽17)

400000
040000
0000160
0001015
0016000
0000016
,
010000
1600000
00161600
002100
00011615
001011
,
14140000
1430000
000011
00111615
00160016
00111616
,
330000
1430000
000011
001616160
00160016
0001601

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,15,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,2,0,1,0,0,16,1,1,0,0,0,0,0,16,1,0,0,0,0,15,1],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,16,0,0,1,15,16,16],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,0,16,16,0,0,0,0,16,0,16,0,0,1,16,0,0,0,0,1,0,16,1] >;

C42.80D4 in GAP, Magma, Sage, TeX

C_4^2._{80}D_4
% in TeX

G:=Group("C4^2.80D4");
// GroupNames label

G:=SmallGroup(128,283);
// by ID

G=gap.SmallGroup(128,283);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,352,1123,1018,248,1971,242]);
// 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=a^2*b^-1*c^3>;
// generators/relations

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