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G = C42.239D4order 128 = 27

221st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.239D4, C42.365C23, C4⋊C4.88C23, (C2×C8).454C23, (C2×C4).333C24, (C22×C4).459D4, C23.388(C2×D4), C4⋊Q8.274C22, (C2×Q8).89C23, C4.98(C4.4D4), (C2×D4).101C23, C8⋊C4.167C22, C41D4.146C22, C23.24D442C2, (C22×C8).460C22, (C2×C42).846C22, C22.5(C4.4D4), C22.593(C22×D4), D4⋊C4.201C22, C2.35(D8⋊C22), C4(C42.29C22), C4(C42.30C22), C4(C42.28C22), (C22×C4).1555C23, C23.37C239C2, Q8⋊C4.202C22, C4.4D4.135C22, C42.C2.111C22, C42.30C2223C2, C42.29C2223C2, C42⋊C2.138C22, C42.28C2237C2, C22.26C24.34C2, (C2×C8⋊C4)⋊39C2, C4.42(C2×C4○D4), (C2×C4).513(C2×D4), C2.44(C2×C4.4D4), (C2×C4).712(C4○D4), (C2×C4○D4).148C22, (C2×C4)(C42.29C22), (C2×C4)(C42.28C22), (C2×C4)(C42.30C22), SmallGroup(128,1867)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.239D4
C1C2C4C2×C4C22×C4C22×C8C2×C8⋊C4 — C42.239D4
C1C2C2×C4 — C42.239D4
C1C2×C4C2×C42 — C42.239D4
C1C2C2C2×C4 — C42.239D4

Generators and relations for C42.239D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2c3 >

Subgroups: 372 in 196 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×8], C8⋊C4 [×4], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42.C2 [×2], C41D4, C4⋊Q8, C4⋊Q8 [×2], C22×C8 [×2], C2×C4○D4 [×2], C2×C8⋊C4, C23.24D4 [×4], C42.28C22 [×4], C42.29C22 [×2], C42.30C22 [×2], C22.26C24, C23.37C23, C42.239D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D8⋊C22 [×2], C42.239D4

Smallest permutation representation of C42.239D4
On 64 points
Generators in S64
(1 64 51 20)(2 61 52 17)(3 58 53 22)(4 63 54 19)(5 60 55 24)(6 57 56 21)(7 62 49 18)(8 59 50 23)(9 44 33 28)(10 41 34 25)(11 46 35 30)(12 43 36 27)(13 48 37 32)(14 45 38 29)(15 42 39 26)(16 47 40 31)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 63 37 59)(34 64 38 60)(35 57 39 61)(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 34 37 14)(10 13 38 33)(11 40 39 12)(15 36 35 16)(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)| (1,64,51,20)(2,61,52,17)(3,58,53,22)(4,63,54,19)(5,60,55,24)(6,57,56,21)(7,62,49,18)(8,59,50,23)(9,44,33,28)(10,41,34,25)(11,46,35,30)(12,43,36,27)(13,48,37,32)(14,45,38,29)(15,42,39,26)(16,47,40,31), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,63,37,59)(34,64,38,60)(35,57,39,61)(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)(9,34,37,14)(10,13,38,33)(11,40,39,12)(15,36,35,16)(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( (1,64,51,20)(2,61,52,17)(3,58,53,22)(4,63,54,19)(5,60,55,24)(6,57,56,21)(7,62,49,18)(8,59,50,23)(9,44,33,28)(10,41,34,25)(11,46,35,30)(12,43,36,27)(13,48,37,32)(14,45,38,29)(15,42,39,26)(16,47,40,31), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,63,37,59)(34,64,38,60)(35,57,39,61)(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)(9,34,37,14)(10,13,38,33)(11,40,39,12)(15,36,35,16)(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([(1,64,51,20),(2,61,52,17),(3,58,53,22),(4,63,54,19),(5,60,55,24),(6,57,56,21),(7,62,49,18),(8,59,50,23),(9,44,33,28),(10,41,34,25),(11,46,35,30),(12,43,36,27),(13,48,37,32),(14,45,38,29),(15,42,39,26),(16,47,40,31)], [(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,63,37,59),(34,64,38,60),(35,57,39,61),(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),(9,34,37,14),(10,13,38,33),(11,40,39,12),(15,36,35,16),(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 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K···4P8A···8H
order1222222244444444444···48···8
size1111228811112244448···84···4

32 irreducible representations

dim111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4D8⋊C22
kernelC42.239D4C2×C8⋊C4C23.24D4C42.28C22C42.29C22C42.30C22C22.26C24C23.37C23C42C22×C4C2×C4C2
# reps114422112284

Matrix representation of C42.239D4 in GL6(𝔽17)

1690000
1310000
0010055
00010125
0012570
00121207
,
100000
010000
004000
000400
000040
000004
,
1320000
140000
00512100
0055010
00010125
00701212
,
1320000
040000
0012570
0055010
0010055
0007512

G:=sub<GL(6,GF(17))| [16,13,0,0,0,0,9,1,0,0,0,0,0,0,10,0,12,12,0,0,0,10,5,12,0,0,5,12,7,0,0,0,5,5,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,5,5,0,7,0,0,12,5,10,0,0,0,10,0,12,12,0,0,0,10,5,12],[13,0,0,0,0,0,2,4,0,0,0,0,0,0,12,5,10,0,0,0,5,5,0,7,0,0,7,0,5,5,0,0,0,10,5,12] >;

C42.239D4 in GAP, Magma, Sage, TeX

C_4^2._{239}D_4
% in TeX

G:=Group("C4^2.239D4");
// GroupNames label

G:=SmallGroup(128,1867);
// by ID

G=gap.SmallGroup(128,1867);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,723,100,248,2804,172,4037,124]);
// Polycyclic

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

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