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G = C42.256D4order 128 = 27

238th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.256D4, C42.384C23, C43(C8⋊Q8), C8⋊Q838C2, C43(C8⋊D4), (C4×Q16)⋊29C2, C8.7(C4○D4), C8⋊D4.4C2, (C4×SD16)⋊16C2, C42(C8.2D4), C42(C8.D4), C8.2D425C2, C8.D436C2, (C4×M4(2))⋊8C2, C4⋊C4.111C23, (C2×C4).370C24, (C2×C8).277C23, (C4×C8).182C22, (C4×D4).91C22, (C22×C4).476D4, C23.266(C2×D4), C4⋊Q8.293C22, (C4×Q8).88C22, (C2×D4).125C23, (C2×Q8).113C23, C2.D8.219C22, C4.Q8.135C22, C8⋊C4.127C22, C4⋊D4.174C22, C4.144(C8.C22), (C2×C42).863C22, (C2×Q16).158C22, C22.630(C22×D4), C22⋊Q8.179C22, D4⋊C4.203C22, C2.43(D8⋊C22), (C22×C4).1050C23, Q8⋊C4.205C22, (C2×SD16).117C22, C4.4D4.145C22, C42.C2.122C22, C42(C42.30C22), C43(C42.28C22), C23.37C2313C2, C42.28C2239C2, C42.30C2224C2, (C2×M4(2)).280C22, C23.36C23.22C2, C2.67(C22.26C24), C4.55(C2×C4○D4), (C2×C4).523(C2×D4), C2.45(C2×C8.C22), SmallGroup(128,1904)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.256D4
C1C2C4C2×C4C42C4×C8C4×M4(2) — C42.256D4
C1C2C2×C4 — C42.256D4
C1C2×C4C2×C42 — C42.256D4
C1C2C2C2×C4 — C42.256D4

Generators and relations for C42.256D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c3 >

Subgroups: 316 in 182 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×4], M4(2) [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C422C2, C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C4×M4(2), C4×SD16 [×2], C4×Q16 [×2], C8⋊D4 [×2], C8.D4 [×2], C42.28C22, C42.30C22, C8.2D4, C8⋊Q8, C23.36C23, C23.37C23, C42.256D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8.C22, D8⋊C22, C42.256D4

Smallest permutation representation of C42.256D4
On 64 points
Generators in S64
(1 46 5 42)(2 43 6 47)(3 48 7 44)(4 45 8 41)(9 53 13 49)(10 50 14 54)(11 55 15 51)(12 52 16 56)(17 34 21 38)(18 39 22 35)(19 36 23 40)(20 33 24 37)(25 63 29 59)(26 60 30 64)(27 57 31 61)(28 62 32 58)
(1 40 27 16)(2 33 28 9)(3 34 29 10)(4 35 30 11)(5 36 31 12)(6 37 32 13)(7 38 25 14)(8 39 26 15)(17 63 54 44)(18 64 55 45)(19 57 56 46)(20 58 49 47)(21 59 50 48)(22 60 51 41)(23 61 52 42)(24 62 53 43)
(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 3)(2 6)(5 7)(10 12)(11 15)(14 16)(17 52)(18 55)(19 50)(20 53)(21 56)(22 51)(23 54)(24 49)(25 31)(27 29)(28 32)(34 36)(35 39)(38 40)(41 64)(42 59)(43 62)(44 57)(45 60)(46 63)(47 58)(48 61)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L···4R8A···8H
order122222444444444444···48···8
size111148111122224448···84···4

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C8.C22D8⋊C22
kernelC42.256D4C4×M4(2)C4×SD16C4×Q16C8⋊D4C8.D4C42.28C22C42.30C22C8.2D4C8⋊Q8C23.36C23C23.37C23C42C22×C4C8C4C2
# reps11222211111122822

Matrix representation of C42.256D4 in GL6(𝔽17)

640000
4110000
001631515
0014102
005548
000121413
,
400000
040000
000400
0013000
001313139
004044
,
100000
010000
000010
0016161615
000100
001001
,
100000
14160000
0001600
0016000
000010
00001616

G:=sub<GL(6,GF(17))| [6,4,0,0,0,0,4,11,0,0,0,0,0,0,16,14,5,0,0,0,3,1,5,12,0,0,15,0,4,14,0,0,15,2,8,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,13,13,4,0,0,4,0,13,0,0,0,0,0,13,4,0,0,0,0,9,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,1,16,0,0,0,0,0,15,0,1],[1,14,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C42.256D4 in GAP, Magma, Sage, TeX

C_4^2._{256}D_4
% in TeX

G:=Group("C4^2.256D4");
// GroupNames label

G:=SmallGroup(128,1904);
// by ID

G=gap.SmallGroup(128,1904);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,184,80,4037,124]);
// 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,b*c=c*b,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations

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