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G = C42.448D4order 128 = 27

81st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.448D4, C42.335C23, C4○D44Q8, D43(C2×Q8), Q83(C2×Q8), Q8⋊Q82C2, D42Q82C2, C4.Q1619C2, D4⋊Q819C2, C4⋊C8.43C22, C4⋊C4.42C23, (C2×C8).26C23, C4.30(C22×Q8), C4⋊M4(2)⋊8C2, (C2×C4).277C24, C4.Q8.9C22, (C22×C4).432D4, C23.659(C2×D4), C4⋊Q8.262C22, C4.89(C22⋊Q8), C4.124(C8⋊C22), C2.D8.80C22, (C4×D4).317C22, (C2×D4).395C23, (C2×Q8).366C23, (C4×Q8).298C22, M4(2)⋊C417C2, D4⋊C4.24C22, C4.118(C8.C22), (C2×C42).823C22, (C22×C4).996C23, Q8⋊C4.25C22, C23.36D4.1C2, C22.537(C22×D4), C22.54(C22⋊Q8), (C2×M4(2)).66C22, C42⋊C2.314C22, (C2×C4⋊Q8)⋊33C2, C4.87(C2×C4○D4), (C4×C4○D4).24C2, C2.24(C2×C8⋊C22), (C2×C4).101(C2×Q8), C2.58(C2×C22⋊Q8), (C2×C4).1438(C2×D4), C2.24(C2×C8.C22), (C2×C4).294(C4○D4), (C2×C4⋊C4).603C22, (C2×C4○D4).308C22, SmallGroup(128,1811)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.448D4
C1C2C4C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.448D4
C1C2C2×C4 — C42.448D4
C1C22C2×C42 — C42.448D4
C1C2C2C2×C4 — C42.448D4

Generators and relations for C42.448D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 372 in 208 conjugacy classes, 104 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×23], D4 [×2], D4 [×5], Q8 [×2], Q8 [×9], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×8], 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, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4, C23.36D4 [×2], C4⋊M4(2), M4(2)⋊C4 [×2], D4⋊Q8 [×2], Q8⋊Q8 [×2], D42Q8 [×2], C4.Q16 [×2], C4×C4○D4, C2×C4⋊Q8, C42.448D4
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], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8⋊C22, C2×C8.C22, C42.448D4

Smallest permutation representation of C42.448D4
On 64 points
Generators in S64
(1 11 41 63)(2 64 42 12)(3 13 43 57)(4 58 44 14)(5 15 45 59)(6 60 46 16)(7 9 47 61)(8 62 48 10)(17 55 39 30)(18 31 40 56)(19 49 33 32)(20 25 34 50)(21 51 35 26)(22 27 36 52)(23 53 37 28)(24 29 38 54)
(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)| (1,11,41,63)(2,64,42,12)(3,13,43,57)(4,58,44,14)(5,15,45,59)(6,60,46,16)(7,9,47,61)(8,62,48,10)(17,55,39,30)(18,31,40,56)(19,49,33,32)(20,25,34,50)(21,51,35,26)(22,27,36,52)(23,53,37,28)(24,29,38,54), (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( (1,11,41,63)(2,64,42,12)(3,13,43,57)(4,58,44,14)(5,15,45,59)(6,60,46,16)(7,9,47,61)(8,62,48,10)(17,55,39,30)(18,31,40,56)(19,49,33,32)(20,25,34,50)(21,51,35,26)(22,27,36,52)(23,53,37,28)(24,29,38,54), (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([(1,11,41,63),(2,64,42,12),(3,13,43,57),(4,58,44,14),(5,15,45,59),(6,60,46,16),(7,9,47,61),(8,62,48,10),(17,55,39,30),(18,31,40,56),(19,49,33,32),(20,25,34,50),(21,51,35,26),(22,27,36,52),(23,53,37,28),(24,29,38,54)], [(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

dim1111111111222244
type++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2D4D4Q8C4○D4C8⋊C22C8.C22
kernelC42.448D4C23.36D4C4⋊M4(2)M4(2)⋊C4D4⋊Q8Q8⋊Q8D42Q8C4.Q16C4×C4○D4C2×C4⋊Q8C42C22×C4C4○D4C2×C4C4C4
# reps1212222211224422

Matrix representation of C42.448D4 in GL6(𝔽17)

1300000
440000
0000160
00161612
001000
00161601
,
100000
010000
000010
00111615
0016000
0011016
,
120000
16160000
0012121010
005507
0051200
0012050
,
16150000
110000
0012121010
001212010
0051200
00120510

G:=sub<GL(6,GF(17))| [13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,16,1,16,0,0,0,16,0,16,0,0,16,1,0,0,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,0,0,0,0,15,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,12,5,5,12,0,0,12,5,12,0,0,0,10,0,0,5,0,0,10,7,0,0],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,12,12,5,12,0,0,12,12,12,0,0,0,10,0,0,5,0,0,10,10,0,10] >;

C42.448D4 in GAP, Magma, Sage, TeX

C_4^2._{448}D_4
% in TeX

G:=Group("C4^2.448D4");
// GroupNames label

G:=SmallGroup(128,1811);
// by ID

G=gap.SmallGroup(128,1811);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,2019,248,4037,1027,124]);
// Polycyclic

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

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