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G = C42.247D4order 128 = 27

229th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.247D4, C42.372C23, (C2×C8)⋊16D4, C4(C83D4), C8.28(C2×D4), C83D424C2, C4(C8.2D4), C4.8(C22×D4), C8.2D424C2, C4.56(C41D4), (C2×C4).348C24, (C2×C8).266C23, C23.390(C2×D4), (C22×C4).467D4, C4⋊Q8.280C22, (C2×D8).162C22, (C2×D4).114C23, C22.3(C41D4), (C2×Q8).102C23, C8⋊C4.117C22, C41D4.152C22, (C22×C8).270C22, (C2×C42).854C22, C22.26C249C2, (C2×Q16).157C22, C22.608(C22×D4), C2.38(D8⋊C22), (C22×C4).1563C23, (C2×SD16).115C22, C4.4D4.141C22, (C2×C4○D8)⋊20C2, (C2×C4)(C83D4), (C2×C8⋊C4)⋊10C2, (C2×C4)(C8.2D4), (C2×C4).519(C2×D4), C2.27(C2×C41D4), (C2×C4○D4).154C22, SmallGroup(128,1882)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.247D4
C1C2C22C2×C4C22×C4C2×C42C2×C8⋊C4 — C42.247D4
C1C2C2×C4 — C42.247D4
C1C2×C4C2×C42 — C42.247D4
C1C2C2C2×C4 — C42.247D4

Generators and relations for C42.247D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 564 in 282 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×24], Q8 [×8], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×4], C4○D4 [×16], C8⋊C4 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C4○D8 [×16], C2×C4○D4 [×4], C2×C8⋊C4, C83D4 [×4], C8.2D4 [×4], C22.26C24 [×2], C2×C4○D8 [×4], C42.247D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], C2×C41D4, D8⋊C22 [×2], C42.247D4

Smallest permutation representation of C42.247D4
On 64 points
Generators in S64
(1 63 55 47)(2 60 56 44)(3 57 49 41)(4 62 50 46)(5 59 51 43)(6 64 52 48)(7 61 53 45)(8 58 54 42)(9 22 40 30)(10 19 33 27)(11 24 34 32)(12 21 35 29)(13 18 36 26)(14 23 37 31)(15 20 38 28)(16 17 39 25)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 41)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)
(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 42 5 46)(2 45 6 41)(3 48 7 44)(4 43 8 47)(9 19 13 23)(10 22 14 18)(11 17 15 21)(12 20 16 24)(25 38 29 34)(26 33 30 37)(27 36 31 40)(28 39 32 35)(49 64 53 60)(50 59 54 63)(51 62 55 58)(52 57 56 61)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A···8H
order1222222222444444444444448···8
size1111228888111122444488884···4

32 irreducible representations

dim1111112224
type+++++++++
imageC1C2C2C2C2C2D4D4D4D8⋊C22
kernelC42.247D4C2×C8⋊C4C83D4C8.2D4C22.26C24C2×C4○D8C42C2×C8C22×C4C2
# reps1144242824

Matrix representation of C42.247D4 in GL6(𝔽17)

010000
1600000
000010
000001
001000
000100
,
100000
010000
004000
000400
000040
000004
,
010000
1600000
0016161414
00116314
003311
00143161
,
100000
0160000
003311
00314116
0016161414
00161143

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,1,3,14,0,0,16,16,3,3,0,0,14,3,1,16,0,0,14,14,1,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,3,3,16,16,0,0,3,14,16,1,0,0,1,1,14,14,0,0,1,16,14,3] >;

C42.247D4 in GAP, Magma, Sage, TeX

C_4^2._{247}D_4
% in TeX

G:=Group("C4^2.247D4");
// GroupNames label

G:=SmallGroup(128,1882);
// by ID

G=gap.SmallGroup(128,1882);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,184,248,2804,172]);
// Polycyclic

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

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