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G = C42.296D4order 128 = 27

278th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.296D4, C4.42- 1+4, C42.430C23, C4.232+ 1+4, Q8.Q83C2, C88D4.1C2, D4⋊Q811C2, Q8⋊Q840C2, C42Q1610C2, Q8.D43C2, D4.D440C2, C4.117(C4○D8), C8.18D419C2, C4⋊C8.340C22, C4⋊C4.187C23, (C2×C8).171C23, (C2×C4).446C24, C23.405(C2×D4), (C22×C4).524D4, C4⋊Q8.325C22, C2.D8.45C22, C4.Q8.93C22, (C4×D4).127C22, D4⋊C4.7C22, (C2×D4).189C23, (C4×Q8).123C22, (C2×Q16).29C22, (C2×Q8).176C23, C4⋊D4.209C22, (C22×C8).189C22, (C2×C42).903C22, (C2×SD16).89C22, C22.706(C22×D4), C22.3(C8.C22), C22⋊Q8.213C22, (C22×C4).1579C23, Q8⋊C4.109C22, C4.4D4.164C22, C42.C2.141C22, C23.37C2325C2, C23.36C23.30C2, C2.65(C22.31C24), (C2×C4⋊C8)⋊31C2, C2.50(C2×C4○D8), (C2×C4).570(C2×D4), C2.66(C2×C8.C22), SmallGroup(128,1980)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.296D4
C1C2C4C2×C4C42C4×D4C23.36C23 — C42.296D4
C1C2C2×C4 — C42.296D4
C1C22C2×C42 — C42.296D4
C1C2C2C2×C4 — C42.296D4

Generators and relations for C42.296D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=c3 >

Subgroups: 316 in 177 conjugacy classes, 88 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C422C2, C4⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×2], C2×Q16 [×2], C2×C4⋊C8, D4.D4, C42Q16, Q8.D4 [×2], C88D4 [×2], C8.18D4 [×2], D4⋊Q8, Q8⋊Q8, Q8.Q8 [×2], C23.36C23, C23.37C23, C42.296D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C4○D8, C2×C8.C22, C42.296D4

Smallest permutation representation of C42.296D4
On 64 points
Generators in S64
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 33 13 37)(10 34 14 38)(11 35 15 39)(12 36 16 40)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)(41 54 45 50)(42 55 46 51)(43 56 47 52)(44 49 48 53)
(1 30 51 10)(2 11 52 31)(3 32 53 12)(4 13 54 25)(5 26 55 14)(6 15 56 27)(7 28 49 16)(8 9 50 29)(17 46 38 59)(18 60 39 47)(19 48 40 61)(20 62 33 41)(21 42 34 63)(22 64 35 43)(23 44 36 57)(24 58 37 45)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 33 21 37)(18 36 22 40)(19 39 23 35)(20 34 24 38)(25 26 29 30)(27 32 31 28)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I4J4K···4Q8A···8H
order12222224···4444···48···8
size11112282···2448···84···4

32 irreducible representations

dim111111111111222444
type+++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+42- 1+4C8.C22
kernelC42.296D4C2×C4⋊C8D4.D4C42Q16Q8.D4C88D4C8.18D4D4⋊Q8Q8⋊Q8Q8.Q8C23.36C23C23.37C23C42C22×C4C4C4C4C22
# reps111122211211228112

Matrix representation of C42.296D4 in GL6(𝔽17)

400000
040000
000010
000001
001000
000100
,
100000
010000
0016200
0016100
0000162
0000161
,
770000
500000
000062
0000711
006200
0071100
,
770000
5100000
000062
0000711
00111500
0010600

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[7,5,0,0,0,0,7,0,0,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,6,7,0,0,0,0,2,11,0,0],[7,5,0,0,0,0,7,10,0,0,0,0,0,0,0,0,11,10,0,0,0,0,15,6,0,0,6,7,0,0,0,0,2,11,0,0] >;

C42.296D4 in GAP, Magma, Sage, TeX

C_4^2._{296}D_4
% in TeX

G:=Group("C4^2.296D4");
// GroupNames label

G:=SmallGroup(128,1980);
// by ID

G=gap.SmallGroup(128,1980);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,80,4037,1027,124]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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