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G = C42.98D4order 128 = 27

80th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.98D4, D41(C4⋊C4), (C4×D4)⋊13C4, (C2×C4).133D8, (C2×D4).22Q8, C43(D4⋊C4), C429C41C2, (C2×D4).198D4, C2.1(C4⋊D8), C22.31(C2×D8), C42.136(C2×C4), C22.4Q161C2, C2.1(D42Q8), C2.1(D4⋊Q8), (C2×C4).103SD16, C23.740(C2×D4), (C22×C4).266D4, C4.93(C22⋊Q8), C4.118(C4⋊D4), C2.1(D4.D4), (C22×C8).10C22, C22.46(C2×SD16), C4.31(C42⋊C2), C22.59(C8⋊C22), (C2×C42).247C22, C22.67(C22⋊Q8), C22.105(C4⋊D4), (C22×C4).1324C23, (C22×D4).456C22, C22.48(C8.C22), C2.11(C23.7Q8), C2.21(C23.36D4), C4.1(C2×C4⋊C4), (C2×C4⋊C8)⋊11C2, (C2×C4×D4).15C2, C4⋊C4.189(C2×C4), (C2×C4).259(C2×Q8), (C2×D4⋊C4).1C2, (C2×D4).205(C2×C4), (C2×C4).1314(C2×D4), C2.18(C2×D4⋊C4), (C2×C4⋊C4).32C22, (C2×C4).860(C4○D4), (C2×C4).362(C22×C4), (C2×C4).358(C22⋊C4), C22.247(C2×C22⋊C4), SmallGroup(128,534)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.98D4
C1C2C22C2×C4C22×C4C22×D4C2×C4×D4 — C42.98D4
C1C2C2×C4 — C42.98D4
C1C23C2×C42 — C42.98D4
C1C2C2C22×C4 — C42.98D4

Generators and relations for C42.98D4
 G = < a,b,c,d | a4=b4=c4=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=a-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 404 in 186 conjugacy classes, 76 normal (36 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×4], C4 [×6], C22 [×7], C22 [×16], C8 [×2], C2×C4 [×6], C2×C4 [×8], C2×C4 [×20], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×6], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, D4⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×4], C4×D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22.4Q16 [×2], C429C4, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C4×D4, C42.98D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.7Q8, C2×D4⋊C4, C23.36D4, C4⋊D8, D4.D4, D4⋊Q8, D42Q8, C42.98D4

Smallest permutation representation of C42.98D4
On 64 points
Generators in S64
(1 63 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 32 35 20)(10 21 36 25)(11 26 37 22)(12 23 38 27)(13 28 39 24)(14 17 40 29)(15 30 33 18)(16 19 34 31)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 12 39 54)(2 53 40 11)(3 10 33 52)(4 51 34 9)(5 16 35 50)(6 49 36 15)(7 14 37 56)(8 55 38 13)(17 26 64 45)(18 44 57 25)(19 32 58 43)(20 42 59 31)(21 30 60 41)(22 48 61 29)(23 28 62 47)(24 46 63 27)
(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)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4N4O4P4Q4R8A···8H
order12···222224···44···444448···8
size11···144442···24···488884···4

38 irreducible representations

dim1111111222222244
type+++++++++-++-
imageC1C2C2C2C2C2C4D4D4D4Q8D8SD16C4○D4C8⋊C22C8.C22
kernelC42.98D4C22.4Q16C429C4C2×D4⋊C4C2×C4⋊C8C2×C4×D4C4×D4C42C22×C4C2×D4C2×D4C2×C4C2×C4C2×C4C22C22
# reps1212118222244411

Matrix representation of C42.98D4 in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000130
000074
,
0160000
100000
000100
0016000
000010
000001
,
1250000
550000
0031400
00141400
000062
0000711
,
1250000
12120000
0031400
003300
00001115
000096

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,7,0,0,0,0,0,4],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,6,7,0,0,0,0,2,11],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,11,9,0,0,0,0,15,6] >;

C42.98D4 in GAP, Magma, Sage, TeX

C_4^2._{98}D_4
% in TeX

G:=Group("C4^2.98D4");
// GroupNames label

G:=SmallGroup(128,534);
// by ID

G=gap.SmallGroup(128,534);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,718,172]);
// Polycyclic

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

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