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G = C42.244D4order 128 = 27

226th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.244D4, C42.369C23, C4⋊C4.95C23, (C4×M4(2))⋊43C2, (C4×C8).353C22, (C2×C4).340C24, (C2×C8).461C23, C23.682(C2×D4), (C22×C4).464D4, C4⋊Q8.277C22, (C2×Q8).95C23, C4.62(C4.4D4), (C2×D4).107C23, C8⋊C4.171C22, C41D4.149C22, C23.36D446C2, (C2×C42).851C22, C22.600(C22×D4), D4⋊C4.135C22, C2.37(D8⋊C22), (C22×C4).1038C23, Q8⋊C4.127C22, C4.4D4.138C22, C22.42(C4.4D4), C42.C2.113C22, C42.29C2219C2, C42.78C2228C2, C42.30C2219C2, (C2×M4(2)).377C22, C22.26C24.36C2, C4.49(C2×C4○D4), (C2×C4).518(C2×D4), (C2×C42.C2)⋊35C2, C2.51(C2×C4.4D4), (C2×C4).304(C4○D4), (C2×C4⋊C4).627C22, (C2×C4○D4).152C22, SmallGroup(128,1874)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.244D4
C1C2C4C2×C4C22×C4C2×M4(2)C4×M4(2) — C42.244D4
C1C2C2×C4 — C42.244D4
C1C22C2×C42 — C42.244D4
C1C2C2C2×C4 — C42.244D4

Generators and relations for C42.244D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >

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 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C42.C2 [×4], C42.C2 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C4×M4(2), C23.36D4 [×4], C42.78C22 [×4], C42.29C22 [×2], C42.30C22 [×2], C2×C42.C2, C22.26C24, C42.244D4
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], C42.244D4

Smallest permutation representation of C42.244D4
On 64 points
Generators in S64
(1 64 51 16)(2 57 52 9)(3 58 53 10)(4 59 54 11)(5 60 55 12)(6 61 56 13)(7 62 49 14)(8 63 50 15)(17 33 31 46)(18 34 32 47)(19 35 25 48)(20 36 26 41)(21 37 27 42)(22 38 28 43)(23 39 29 44)(24 40 30 45)
(1 22 5 18)(2 19 6 23)(3 24 7 20)(4 21 8 17)(9 48 13 44)(10 45 14 41)(11 42 15 46)(12 47 16 43)(25 56 29 52)(26 53 30 49)(27 50 31 54)(28 55 32 51)(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 54 51 4)(2 3 52 53)(5 50 55 8)(6 7 56 49)(9 62 57 14)(10 13 58 61)(11 60 59 12)(15 64 63 16)(17 22 31 28)(18 27 32 21)(19 20 25 26)(23 24 29 30)(33 47 46 34)(35 45 48 40)(36 39 41 44)(37 43 42 38)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4D8⋊C22
kernelC42.244D4C4×M4(2)C23.36D4C42.78C22C42.29C22C42.30C22C2×C42.C2C22.26C24C42C22×C4C2×C4C2
# reps114422112284

Matrix representation of C42.244D4 in GL6(𝔽17)

1680000
410000
0013000
0001300
0000130
0000013
,
1600000
0160000
000010
000001
0016000
0001600
,
420000
1130000
00152215
00151522
00215215
002222
,
420000
0130000
00215152
00151522
00152152
002222

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

C42.244D4 in GAP, Magma, Sage, TeX

C_4^2._{244}D_4
% in TeX

G:=Group("C4^2.244D4");
// GroupNames label

G:=SmallGroup(128,1874);
// by ID

G=gap.SmallGroup(128,1874);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,1018,521,248,2804,172,4037,124]);
// 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*b^2,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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