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G = C42.280D4order 128 = 27

262nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.280D4, C42.732C23, C4.522- 1+4, D4.Q81C2, C82Q813C2, C83Q821C2, C4.30(C4○D8), C4.4D814C2, (C4×C8).77C22, C4⋊C4.167C23, C4⋊C8.313C22, C4.29(C8⋊C22), (C2×C8).166C23, (C2×C4).426C24, C23.296(C2×D4), (C22×C4).509D4, C4⋊Q8.310C22, C4.Q8.84C22, C2.D8.36C22, (C2×D4).175C23, (C4×D4).113C22, C23.19D43C2, C42.12C437C2, C41D4.171C22, C4⋊D4.198C22, C22⋊C8.195C22, (C2×C42).887C22, C22.686(C22×D4), D4⋊C4.111C22, (C22×C4).1091C23, C42.C2.129C22, C23.37C2320C2, C42⋊C2.163C22, C22.26C24.45C2, C2.74(C23.38C23), C2.45(C2×C4○D8), (C2×C4).553(C2×D4), C2.60(C2×C8⋊C22), SmallGroup(128,1960)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.280D4
C1C2C4C2×C4C22×C4C42⋊C2C23.37C23 — C42.280D4
C1C2C2×C4 — C42.280D4
C1C22C2×C42 — C42.280D4
C1C2C2C2×C4 — C42.280D4

Generators and relations for C42.280D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=a2b2c3 >

Subgroups: 364 in 185 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×6], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×3], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4, C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C4.4D4, C42.C2 [×2], C41D4, C4⋊Q8 [×3], C2×C4○D4, C42.12C4, D4.Q8 [×4], C23.19D4 [×4], C4.4D8 [×2], C83Q8, C82Q8, C22.26C24, C23.37C23, C42.280D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C2×C4○D8, C2×C8⋊C22, C42.280D4

Smallest permutation representation of C42.280D4
On 64 points
Generators in S64
(1 61 29 41)(2 62 30 42)(3 63 31 43)(4 64 32 44)(5 57 25 45)(6 58 26 46)(7 59 27 47)(8 60 28 48)(9 20 37 56)(10 21 38 49)(11 22 39 50)(12 23 40 51)(13 24 33 52)(14 17 34 53)(15 18 35 54)(16 19 36 55)
(1 39 5 35)(2 12 6 16)(3 33 7 37)(4 14 8 10)(9 31 13 27)(11 25 15 29)(17 60 21 64)(18 41 22 45)(19 62 23 58)(20 43 24 47)(26 36 30 40)(28 38 32 34)(42 51 46 55)(44 53 48 49)(50 57 54 61)(52 59 56 63)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 33)(10 12)(11 39)(13 37)(14 16)(15 35)(17 55)(19 53)(20 24)(21 51)(23 49)(27 31)(34 36)(38 40)(41 61)(42 48)(43 59)(44 46)(45 57)(47 63)(52 56)(58 64)(60 62)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4J4K4L···4Q8A···8H
order12222224···444···48···8
size11114882···248···84···4

32 irreducible representations

dim11111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D8C8⋊C222- 1+4
kernelC42.280D4C42.12C4D4.Q8C23.19D4C4.4D8C83Q8C82Q8C22.26C24C23.37C23C42C22×C4C4C4C4
# reps11442111122822

Matrix representation of C42.280D4 in GL6(𝔽17)

100000
010000
0001600
001000
00116162
0010161
,
1300000
0130000
000010
00161115
0016000
00161016
,
330000
1430000
0051200
005500
0012507
00121057
,
010000
100000
001000
0001600
0000160
00016161

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,0,0,16,0,16,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,16,16,0,0,0,1,0,1,0,0,1,1,0,0,0,0,0,15,0,16],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,5,5,12,12,0,0,12,5,5,10,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,1] >;

C42.280D4 in GAP, Magma, Sage, TeX

C_4^2._{280}D_4
% in TeX

G:=Group("C4^2.280D4");
// GroupNames label

G:=SmallGroup(128,1960);
// by ID

G=gap.SmallGroup(128,1960);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,248,4037,1027,124]);
// Polycyclic

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

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