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G = C42.297D4order 128 = 27

279th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.297D4, C4.52- 1+4, C42.431C23, C4.242+ 1+4, (C2×C4)⋊5Q16, C4.49(C2×Q16), C4.Q1611C2, C42Q1611C2, C22.4(C2×Q16), C4⋊C4.188C23, C4⋊C8.293C22, (C2×C8).172C23, (C2×C4).447C24, C8.18D4.7C2, (C22×C4).525D4, C23.406(C2×D4), C4⋊Q8.326C22, C2.17(C22×Q16), C2.D8.46C22, Q8⋊C4.9C22, (C2×Q8).177C23, (C2×Q16).30C22, (C4×Q8).124C22, (C22×C8).154C22, (C2×C42).904C22, C22.707(C22×D4), C22⋊Q8.214C22, C2.70(D8⋊C22), (C22×C4).1580C23, C23.37C23.43C2, C2.66(C22.31C24), (C2×C4⋊C8).43C2, (C2×C4).571(C2×D4), SmallGroup(128,1981)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.297D4
C1C2C4C2×C4C42C4×Q8C23.37C23 — C42.297D4
C1C2C2×C4 — C42.297D4
C1C22C2×C42 — C42.297D4
C1C2C2C2×C4 — C42.297D4

Generators and relations for C42.297D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2c3 >

Subgroups: 292 in 174 conjugacy classes, 94 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], Q8 [×12], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C2×C8 [×2], Q16 [×4], C22×C4 [×3], C2×Q8 [×4], C2×Q8 [×2], Q8⋊C4 [×8], C4⋊C8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×4], C22×C8 [×2], C2×Q16 [×4], C2×C4⋊C8, C42Q16 [×4], C8.18D4 [×4], C4.Q16 [×4], C23.37C23 [×2], C42.297D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C22×Q16, D8⋊C22, C42.297D4

Smallest permutation representation of C42.297D4
On 64 points
Generators in S64
(1 7 5 3)(2 8 6 4)(9 38 13 34)(10 39 14 35)(11 40 15 36)(12 33 16 37)(17 23 21 19)(18 24 22 20)(25 44 29 48)(26 45 30 41)(27 46 31 42)(28 47 32 43)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 52 63 17)(2 22 64 49)(3 54 57 19)(4 24 58 51)(5 56 59 21)(6 18 60 53)(7 50 61 23)(8 20 62 55)(9 47 36 30)(10 27 37 44)(11 41 38 32)(12 29 39 46)(13 43 40 26)(14 31 33 48)(15 45 34 28)(16 25 35 42)
(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 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim111111222444
type++++++++-+-
imageC1C2C2C2C2C2D4D4Q162+ 1+42- 1+4D8⋊C22
kernelC42.297D4C2×C4⋊C8C42Q16C8.18D4C4.Q16C23.37C23C42C22×C4C2×C4C4C4C2
# reps114442228112

Matrix representation of C42.297D4 in GL6(𝔽17)

400000
13130000
004000
000400
000040
000004
,
400000
13130000
0041300
0081300
0000134
000094
,
1500000
580000
008000
0016900
0000150
0000132
,
910000
380000
000020
0000415
008000
0016900

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

C42.297D4 in GAP, Magma, Sage, TeX

C_4^2._{297}D_4
% in TeX

G:=Group("C4^2.297D4");
// GroupNames label

G:=SmallGroup(128,1981);
// by ID

G=gap.SmallGroup(128,1981);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,352,675,80,4037,1027,124]);
// Polycyclic

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

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