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G = C42.261D4order 128 = 27

243rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.261D4, C42.726C23, C83(C4○D4), C82D48C2, C83Q87C2, C84D420C2, D8⋊C418C2, C4.4D844C2, C4⋊C4.123C23, C4.24(C8⋊C22), (C4×M4(2))⋊11C2, (C4×C8).187C22, (C2×C8).463C23, (C2×C4).382C24, (C2×D8).67C22, C23.269(C2×D4), (C22×C4).480D4, C4⋊Q8.297C22, C4.Q8.33C22, (C2×D4).136C23, (C4×D4).102C22, C8⋊C4.139C22, C41D4.159C22, C4⋊D4.179C22, (C2×C42).868C22, C22.642(C22×D4), D4⋊C4.140C22, (C22×C4).1060C23, C22.26C2415C2, (C2×M4(2)).290C22, C2.79(C22.26C24), C4.67(C2×C4○D4), C2.48(C2×C8⋊C22), (C2×C4).1225(C2×D4), SmallGroup(128,1916)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.261D4
C1C2C4C2×C4C42C8⋊C4C4×M4(2) — C42.261D4
C1C2C2×C4 — C42.261D4
C1C22C2×C42 — C42.261D4
C1C2C2C2×C4 — C42.261D4

Generators and relations for C42.261D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=a2b, bd=db, dcd=a2c3 >

Subgroups: 484 in 223 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×6], C4 [×7], C22, C22 [×15], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×24], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], C4.Q8 [×4], C2×C42, C4×D4 [×4], C4×D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×D8 [×4], C2×C4○D4 [×2], C4×M4(2), D8⋊C4 [×4], C82D4 [×4], C4.4D8 [×2], C84D4, C83Q8, C22.26C24 [×2], C42.261D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8⋊C22 [×2], C42.261D4

Smallest permutation representation of C42.261D4
On 64 points
Generators in S64
(1 49 5 53)(2 54 6 50)(3 51 7 55)(4 56 8 52)(9 44 13 48)(10 41 14 45)(11 46 15 42)(12 43 16 47)(17 35 21 39)(18 40 22 36)(19 37 23 33)(20 34 24 38)(25 59 29 63)(26 64 30 60)(27 61 31 57)(28 58 32 62)
(1 40 31 10)(2 37 32 15)(3 34 25 12)(4 39 26 9)(5 36 27 14)(6 33 28 11)(7 38 29 16)(8 35 30 13)(17 64 44 56)(18 61 45 53)(19 58 46 50)(20 63 47 55)(21 60 48 52)(22 57 41 49)(23 62 42 54)(24 59 43 51)
(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)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 51)(34 50)(35 49)(36 56)(37 55)(38 54)(39 53)(40 52)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4J4K4L4M4N4O8A···8H
order1222222224···4444448···8
size1111488882···2488884···4

32 irreducible representations

dim111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22
kernelC42.261D4C4×M4(2)D8⋊C4C82D4C4.4D8C84D4C83Q8C22.26C24C42C22×C4C8C4
# reps114421122284

Matrix representation of C42.261D4 in GL6(𝔽17)

0130000
400000
0000160
00134162
001000
0000013
,
400000
040000
00413115
0000160
0001600
00160413
,
010000
1600000
0051206
00125611
0031400
0031157
,
100000
0160000
00512116
00125011
00141400
0031157

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,1,0,0,0,0,4,0,0,0,0,16,16,0,0,0,0,0,2,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,16,0,0,13,0,16,0,0,0,1,16,0,4,0,0,15,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,5,12,3,3,0,0,12,5,14,11,0,0,0,6,0,5,0,0,6,11,0,7],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,5,12,14,3,0,0,12,5,14,11,0,0,11,0,0,5,0,0,6,11,0,7] >;

C42.261D4 in GAP, Magma, Sage, TeX

C_4^2._{261}D_4
% in TeX

G:=Group("C4^2.261D4");
// GroupNames label

G:=SmallGroup(128,1916);
// by ID

G=gap.SmallGroup(128,1916);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,520,521,80,4037,124]);
// Polycyclic

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

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