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G = C42.243D4order 128 = 27

225th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.243D4, C42.709C23, C4.4D841C2, C4⋊C4.94C23, C4.21(C8⋊C22), (C4×M4(2))⋊42C2, (C4×C8).352C22, (C2×C4).339C24, (C2×C8).460C23, C4.SD1642C2, C23.681(C2×D4), (C22×C4).463D4, C4⋊Q8.276C22, (C2×Q8).94C23, C4.61(C4.4D4), (C2×D4).106C23, C4.21(C8.C22), C8⋊C4.170C22, C41D4.148C22, C23.36D445C2, (C2×C42).850C22, C22.599(C22×D4), D4⋊C4.134C22, (C22×C4).1037C23, Q8⋊C4.126C22, C4.4D4.137C22, C22.41(C4.4D4), C42.28C2232C2, (C2×M4(2)).376C22, C22.26C24.35C2, (C2×C4⋊Q8)⋊36C2, C4.48(C2×C4○D4), (C2×C4).517(C2×D4), C2.40(C2×C8⋊C22), C2.50(C2×C4.4D4), C2.40(C2×C8.C22), (C2×C4).303(C4○D4), (C2×C4⋊C4).626C22, (C2×C4○D4).151C22, SmallGroup(128,1873)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.243D4
C1C2C4C2×C4C22×C4C2×M4(2)C4×M4(2) — C42.243D4
C1C2C2×C4 — C42.243D4
C1C22C2×C42 — C42.243D4
C1C2C2C2×C4 — C42.243D4

Generators and relations for C42.243D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >

Subgroups: 420 in 212 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×12], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×8], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4 [×2], C4×M4(2), C23.36D4 [×4], C4.4D8 [×2], C4.SD16 [×2], C42.28C22 [×4], C2×C4⋊Q8, C22.26C24, C42.243D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C42.243D4

Smallest permutation representation of C42.243D4
On 64 points
Generators in S64
(1 60 55 20)(2 61 56 21)(3 62 49 22)(4 63 50 23)(5 64 51 24)(6 57 52 17)(7 58 53 18)(8 59 54 19)(9 46 35 26)(10 47 36 27)(11 48 37 28)(12 41 38 29)(13 42 39 30)(14 43 40 31)(15 44 33 32)(16 45 34 25)
(1 45 5 41)(2 42 6 46)(3 47 7 43)(4 44 8 48)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)(25 51 29 55)(26 56 30 52)(27 53 31 49)(28 50 32 54)(33 59 37 63)(34 64 38 60)(35 61 39 57)(36 58 40 62)
(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 50 55 4)(2 3 56 49)(5 54 51 8)(6 7 52 53)(9 36 35 10)(11 34 37 16)(12 15 38 33)(13 40 39 14)(17 58 57 18)(19 64 59 24)(20 23 60 63)(21 62 61 22)(25 48 45 28)(26 27 46 47)(29 44 41 32)(30 31 42 43)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim1111111122244
type+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22C8.C22
kernelC42.243D4C4×M4(2)C23.36D4C4.4D8C4.SD16C42.28C22C2×C4⋊Q8C22.26C24C42C22×C4C2×C4C4C4
# reps1142241122822

Matrix representation of C42.243D4 in GL6(𝔽17)

180000
4160000
001000
000100
000010
000001
,
1600000
0160000
000010
00116115
0016000
0016101
,
400000
040000
000001
0010116
00116016
0001600
,
400000
16130000
0000016
0010116
0016101
0016000

G:=sub<GL(6,GF(17))| [1,4,0,0,0,0,8,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,16,0,0,0,1,0,0,0,0,1,16,16,0],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,0,1,16,16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,16,1,0] >;

C42.243D4 in GAP, Magma, Sage, TeX

C_4^2._{243}D_4
% in TeX

G:=Group("C4^2.243D4");
// GroupNames label

G:=SmallGroup(128,1873);
// by ID

G=gap.SmallGroup(128,1873);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,521,248,2804,172,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,a*c=c*a,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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