Copied to
clipboard

G = C42.510C23order 128 = 27

371st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.510C23, C4.312- 1+4, Q824C2, C84Q86C2, C4⋊C4.175D4, Q8⋊Q825C2, C42Q1642C2, (C2×Q8).243D4, D42Q8.1C2, Q16⋊C427C2, C4⋊C4.437C23, C4⋊C8.134C22, (C2×C8).117C23, (C4×C8).297C22, (C2×C4).561C24, Q8.36(C4○D4), C4.SD1636C2, (C4×SD16).16C2, C4⋊Q8.190C22, Q8.D4.2C2, C8⋊C4.60C22, C2.69(Q85D4), (C4×D4).200C22, (C2×D4).272C23, C4.51(C8.C22), (C2×Q16).90C22, (C2×Q8).256C23, (C4×Q8).192C22, C4.Q8.179C22, C2.101(D4○SD16), Q8⋊C4.87C22, C4.4D4.79C22, C22.821(C22×D4), D4⋊C4.210C22, (C2×SD16).173C22, C22.50C24.9C2, C42.28C22.1C2, C4.262(C2×C4○D4), (C2×C4).637(C2×D4), C2.86(C2×C8.C22), SmallGroup(128,2101)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.510C23
C1C2C4C2×C4C42C4×Q8Q82 — C42.510C23
C1C2C2×C4 — C42.510C23
C1C22C4×Q8 — C42.510C23
C1C2C2C2×C4 — C42.510C23

Generators and relations for C42.510C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >

Subgroups: 296 in 169 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×2], Q8 [×2], Q8 [×10], C23, C42, C42 [×2], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], SD16 [×2], Q16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×3], C4×C8, C8⋊C4 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8, C4.Q8 [×2], C42⋊C2, C4×D4, C4×Q8 [×2], C4×Q8 [×4], C4×Q8, C22⋊Q8, C4.4D4 [×2], C422C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C4⋊Q8 [×3], C2×SD16, C2×Q16 [×2], C4×SD16, Q16⋊C4 [×2], C84Q8, C42Q16, Q8.D4 [×2], Q8⋊Q8 [×2], D42Q8, C4.SD16, C42.28C22 [×2], C22.50C24, Q82, C42.510C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8.C22 [×2], C22×D4, C2×C4○D4, 2- 1+4, Q85D4, C2×C8.C22, D4○SD16, C42.510C23

Character table of C42.510C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R8A8B8C8D8E8F
 size 11118222244444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-11111111111-11-11-1-111-1-1-1-1-1-1    linear of order 2
ρ31111-1-11-111-1-1-11-11-111-11-111-11-1-11    linear of order 2
ρ411111-11-111-1-1-11-1-1-1-111-1-11-11-111-1    linear of order 2
ρ51111-1-11-111-1-111-1111-1-111-1-11-111-1    linear of order 2
ρ611111-11-111-1-111-1-11-1-11-11-11-11-1-11    linear of order 2
ρ7111111111111-1111-11-111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-11111111-111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ911111-11-11-11-1-1-11-1-1-11-111-1-11-11-11    linear of order 2
ρ101111-1-11-11-11-1-1-111-1111-11-11-11-11-1    linear of order 2
ρ111111-11111-1-111-1-1-11-1111-1-1-1-1-1-111    linear of order 2
ρ12111111111-1-111-1-11111-1-1-1-11111-1-1    linear of order 2
ρ131111-11111-1-11-1-1-1-1-1-1-111111111-1-1    linear of order 2
ρ14111111111-1-11-1-1-11-11-1-1-111-1-1-1-111    linear of order 2
ρ1511111-11-11-11-11-11-11-1-1-11-111-11-11-1    linear of order 2
ρ161111-1-11-11-11-11-11111-11-1-11-11-11-11    linear of order 2
ρ1722220-2-2-2-22-220-2200000000000000    orthogonal lifted from D4
ρ1822220-2-2-2-2-22202-200000000000000    orthogonal lifted from D4
ρ19222202-22-222-20-2-200000000000000    orthogonal lifted from D4
ρ20222202-22-2-2-2-202200000000000000    orthogonal lifted from D4
ρ212-22-20020-2000-2002i2-2i0000002i0-2i00    complex lifted from C4○D4
ρ222-22-20020-2000200-2i-22i0000002i0-2i00    complex lifted from C4○D4
ρ232-22-20020-20002002i-2-2i000000-2i02i00    complex lifted from C4○D4
ρ242-22-20020-2000-200-2i22i000000-2i02i00    complex lifted from C4○D4
ρ254-4-440-404000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-400-40400000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ274-4-44040-4000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-20-2-2000    complex lifted from D4○SD16

Smallest permutation representation of C42.510C23
On 64 points
Generators in S64
(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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 23)(18 24)(19 21)(20 22)(29 37)(30 38)(31 39)(32 40)(33 43)(34 44)(35 41)(36 42)(45 49)(46 50)(47 51)(48 52)(53 55)(54 56)(57 59)(58 60)
(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)
(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 56 9 60)(6 53 10 57)(7 54 11 58)(8 55 12 59)(13 46 62 52)(14 47 63 49)(15 48 64 50)(16 45 61 51)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)

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

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

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

Matrix representation of C42.510C23 in GL8(𝔽17)

01000000
160000000
00010000
001600000
00000010
00000001
000016000
000001600
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
10000000
01000000
001600000
000160000
00001000
000001600
00000010
000000016
,
005120000
0012120000
512000000
1212000000
00000107
0000160100
000007016
000010010
,
00100000
00010000
10000000
01000000
0000121200
000012500
0000001212
000000125

G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,16,0,10,0,0,0,0,1,0,7,0,0,0,0,0,0,10,0,1,0,0,0,0,7,0,16,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,12,5] >;

C42.510C23 in GAP, Magma, Sage, TeX

C_4^2._{510}C_2^3
% in TeX

G:=Group("C4^2.510C2^3");
// GroupNames label

G:=SmallGroup(128,2101);
// by ID

G=gap.SmallGroup(128,2101);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,723,352,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.510C23 in TeX

׿
×
𝔽