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G = C42.449D4order 128 = 27

82nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.449D4, C42.336C23, C4○D4.9Q8, Q8.4(C2×Q8), D4.4(C2×Q8), D4.Q813C2, Q8.Q813C2, C4⋊C4.43C23, C4⋊C8.44C22, (C2×C8).27C23, C4.31(C22×Q8), C4⋊M4(2)⋊9C2, (C2×C4).278C24, (C22×C4).433D4, C23.660(C2×D4), C4.90(C22⋊Q8), C4.Q8.10C22, C2.D8.81C22, (C2×D4).396C23, (C4×D4).318C22, (C2×Q8).367C23, (C4×Q8).299C22, M4(2)⋊C418C2, D4⋊C4.25C22, (C2×C42).824C22, (C22×C4).997C23, Q8⋊C4.26C22, C23.36D4.2C2, C22.538(C22×D4), C22.55(C22⋊Q8), C2.20(D8⋊C22), (C2×M4(2)).67C22, C42.C2.104C22, C42⋊C2.315C22, C4.88(C2×C4○D4), (C4×C4○D4).25C2, (C2×C4).102(C2×Q8), C2.59(C2×C22⋊Q8), (C2×C4).1215(C2×D4), (C2×C42.C2)⋊32C2, (C2×C4).295(C4○D4), (C2×C4⋊C4).604C22, (C2×C4○D4).309C22, SmallGroup(128,1812)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.449D4
C1C2C4C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.449D4
C1C2C2×C4 — C42.449D4
C1C22C2×C42 — C42.449D4
C1C2C2C2×C4 — C42.449D4

Generators and relations for C42.449D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2c3 >

Subgroups: 324 in 192 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×21], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×2], C2×M4(2) [×2], C2×C4○D4, C23.36D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], D4.Q8 [×4], Q8.Q8 [×4], C4×C4○D4, C2×C42.C2, C42.449D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D8⋊C22 [×2], C42.449D4

Smallest permutation representation of C42.449D4
On 64 points
Generators in S64
(1 15 45 63)(2 60 46 12)(3 9 47 57)(4 62 48 14)(5 11 41 59)(6 64 42 16)(7 13 43 61)(8 58 44 10)(17 55 35 26)(18 31 36 52)(19 49 37 28)(20 25 38 54)(21 51 39 30)(22 27 40 56)(23 53 33 32)(24 29 34 50)
(1 26 5 30)(2 31 6 27)(3 28 7 32)(4 25 8 29)(9 19 13 23)(10 24 14 20)(11 21 15 17)(12 18 16 22)(33 57 37 61)(34 62 38 58)(35 59 39 63)(36 64 40 60)(41 51 45 55)(42 56 46 52)(43 53 47 49)(44 50 48 54)
(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 41 8)(2 7 42 47)(3 46 43 6)(4 5 44 45)(9 64 61 12)(10 11 62 63)(13 60 57 16)(14 15 58 59)(17 38 39 24)(18 23 40 37)(19 36 33 22)(20 21 34 35)(25 26 50 51)(27 32 52 49)(28 56 53 31)(29 30 54 55)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4P4Q4R4S4T8A8B8C8D
order122222224···44···444448888
size111122442···24···488888888

32 irreducible representations

dim1111111122224
type++++++++++-
imageC1C2C2C2C2C2C2C2D4D4Q8C4○D4D8⋊C22
kernelC42.449D4C23.36D4C4⋊M4(2)M4(2)⋊C4D4.Q8Q8.Q8C4×C4○D4C2×C42.C2C42C22×C4C4○D4C2×C4C2
# reps1212441122444

Matrix representation of C42.449D4 in GL6(𝔽17)

4150000
0130000
0000130
0000013
004000
000400
,
1600000
0160000
000010
000001
0016000
0001600
,
400000
16130000
007700
005000
00001010
0000120
,
1300000
140000
007700
0051000
00001010
0000127

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

C42.449D4 in GAP, Magma, Sage, TeX

C_4^2._{449}D_4
% in TeX

G:=Group("C4^2.449D4");
// GroupNames label

G:=SmallGroup(128,1812);
// by ID

G=gap.SmallGroup(128,1812);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,2019,248,4037,1027,124]);
// Polycyclic

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

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