Copied to
clipboard

G = C42.451D4order 128 = 27

84th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.451D4, C42.342C23, (C4×Q16)⋊20C2, (C4×SD16)⋊3C2, C43(D4⋊Q8), C43(Q8⋊Q8), D4⋊Q850C2, Q8⋊Q850C2, C43(C42Q16), C42Q1649C2, D4.4(C4○D4), C4⋊C4.60C23, Q8.3(C4○D4), C43(D4.D4), C43(D4.7D4), D4.D450C2, C4.113(C4○D8), C4⋊C8.335C22, (C2×C4).305C24, (C4×C8).108C22, (C2×C8).317C23, D4.7D4.4C2, (C22×C4).446D4, C23.251(C2×D4), C4⋊Q8.266C22, (C2×Q8).75C23, (C4×Q8).71C22, (C2×D4).402C23, (C4×D4).320C22, C42.12C431C2, C4.Q8.153C22, C2.D8.172C22, C43(C23.20D4), C23.20D454C2, C4.142(C8.C22), C22⋊C8.190C22, (C2×C42).832C22, (C2×Q16).121C22, C22.565(C22×D4), C22⋊Q8.167C22, D4⋊C4.161C22, (C22×C4).1021C23, C23.37C235C2, Q8⋊C4.154C22, (C2×SD16).141C22, C42⋊C2.321C22, C2.106(C22.19C24), C2.26(C2×C4○D8), (C4×C4○D4).26C2, C4.190(C2×C4○D4), (C2×C4).1584(C2×D4), C2.30(C2×C8.C22), (C2×C4○D4).312C22, SmallGroup(128,1839)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.451D4
C1C2C4C2×C4C42C4×D4C4×C4○D4 — C42.451D4
C1C2C2×C4 — C42.451D4
C1C2×C4C2×C42 — C42.451D4
C1C2C2C2×C4 — C42.451D4

Generators and relations for C42.451D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 332 in 195 conjugacy classes, 92 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×21], D4 [×2], D4 [×5], Q8 [×2], Q8 [×7], C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×SD16 [×2], C2×Q16 [×2], C2×C4○D4, C42.12C4, C4×SD16 [×2], C4×Q16 [×2], D4.7D4 [×2], D4.D4, C42Q16, D4⋊Q8, Q8⋊Q8, C23.20D4 [×2], C4×C4○D4, C23.37C23, C42.451D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C4○D8, C2×C8.C22, C42.451D4

Smallest permutation representation of C42.451D4
On 64 points
Generators in S64
(1 22 5 18)(2 64 6 60)(3 24 7 20)(4 58 8 62)(9 25 13 29)(10 43 14 47)(11 27 15 31)(12 45 16 41)(17 50 21 54)(19 52 23 56)(26 34 30 38)(28 36 32 40)(33 46 37 42)(35 48 39 44)(49 61 53 57)(51 63 55 59)
(1 30 55 43)(2 31 56 44)(3 32 49 45)(4 25 50 46)(5 26 51 47)(6 27 52 48)(7 28 53 41)(8 29 54 42)(9 17 33 62)(10 18 34 63)(11 19 35 64)(12 20 36 57)(13 21 37 58)(14 22 38 59)(15 23 39 60)(16 24 40 61)
(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 18 21 22)(19 24 23 20)(25 26 29 30)(27 32 31 28)(33 34 37 38)(35 40 39 36)(41 48 45 44)(42 43 46 47)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4L4M···4S4T4U4V4W8A···8H
order122222244444···44···444448···8
size111144411112···24···488884···4

38 irreducible representations

dim111111111111222224
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4C4○D8C8.C22
kernelC42.451D4C42.12C4C4×SD16C4×Q16D4.7D4D4.D4C42Q16D4⋊Q8Q8⋊Q8C23.20D4C4×C4○D4C23.37C23C42C22×C4D4Q8C4C4
# reps112221111211224482

Matrix representation of C42.451D4 in GL4(𝔽17) generated by

4000
0400
00160
0001
,
4000
0400
0040
0004
,
12500
121200
0001
00160
,
12500
5500
0001
0010
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[12,12,0,0,5,12,0,0,0,0,0,16,0,0,1,0],[12,5,0,0,5,5,0,0,0,0,0,1,0,0,1,0] >;

C42.451D4 in GAP, Magma, Sage, TeX

C_4^2._{451}D_4
% in TeX

G:=Group("C4^2.451D4");
// GroupNames label

G:=SmallGroup(128,1839);
// by ID

G=gap.SmallGroup(128,1839);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,304,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽