Copied to
clipboard

G = C42.75C23order 128 = 27

75th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.75C23, C4.1512+ 1+4, C4⋊C4.187D4, (C4×Q16)⋊35C2, C84Q822C2, C85D4.5C2, Q83Q814C2, C42Q1645C2, C8.21(C4○D4), C8.2D421C2, (C2×Q8).255D4, C4⋊SD16.3C2, D4.D428C2, C4⋊C4.451C23, C4⋊C8.153C22, (C4×C8).210C22, (C2×C4).592C24, (C2×C8).381C23, Q8.D448C2, C4⋊Q8.218C22, SD16⋊C450C2, C8⋊C4.79C22, C2.46(Q86D4), (C4×D4).225C22, (C2×D4).286C23, C4.83(C8.C22), (C2×Q8).271C23, (C2×Q16).95C22, (C4×Q8).215C22, C2.D8.229C22, C2.114(D4○SD16), C41D4.110C22, Q8⋊C4.96C22, (C2×SD16).77C22, C4.4D4.92C22, C22.852(C22×D4), D4⋊C4.100C22, C22.53C24.6C2, C4.170(C2×C4○D4), (C2×C4).656(C2×D4), C2.93(C2×C8.C22), SmallGroup(128,2132)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.75C23
C1C2C4C2×C4C42C4×Q8C22.53C24 — C42.75C23
C1C2C2×C4 — C42.75C23
C1C22C4×Q8 — C42.75C23
C1C2C2C2×C4 — C42.75C23

Generators and relations for C42.75C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b2, e2=a2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

Subgroups: 344 in 181 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×6], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×7], Q8 [×10], C23 [×2], C42, C42 [×2], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], SD16 [×8], Q16 [×4], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×2], C4×C8, C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8, C4×D4 [×2], C4×D4, C4×Q8 [×3], C4×Q8 [×2], C4×Q8, C22.D4 [×2], C4.4D4 [×2], C4.4D4, C42.C2 [×3], C41D4, C4⋊Q8, C4⋊Q8 [×2], C2×SD16 [×6], C2×Q16, C2×Q16 [×2], C4×Q16, SD16⋊C4 [×2], C84Q8, C4⋊SD16, D4.D4 [×2], C42Q16, Q8.D4 [×2], C85D4, C8.2D4 [×2], Q83Q8, C22.53C24, C42.75C23
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, Q86D4, C2×C8.C22, D4○SD16, C42.75C23

Character table of C42.75C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11118822224444444448888444488
ρ111111111111111111111111111111    trivial
ρ211111-1-11-11-11-11-11-11-1-111-11-11-1-11    linear of order 2
ρ311111-1-11-11-11-11-111111-1-1-1-11-111-1    linear of order 2
ρ41111111111111111-11-1-1-1-11-1-1-1-1-1-1    linear of order 2
ρ51111-11-11-11-11-1-1-11-1-1-1-1111-11-111-1    linear of order 2
ρ61111-1-11111111-1111-11111-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-11111111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111-11-11-11-11-1-1-111-111-1-111-11-1-11    linear of order 2
ρ91111-1-11111-1-111-1-1-11-111-11-1-1-1-111    linear of order 2
ρ101111-11-11-111-1-111-1111-11-1-1-11-11-11    linear of order 2
ρ111111-11-11-111-1-111-1-11-11-11-11-11-11-1    linear of order 2
ρ121111-1-11111-1-111-1-1111-1-1111111-1-1    linear of order 2
ρ1311111-1-11-111-1-1-11-11-11-11-111-11-11-1    linear of order 2
ρ141111111111-1-11-1-1-1-1-1-111-1-11111-1-1    linear of order 2
ρ151111111111-1-11-1-1-11-11-1-11-1-1-1-1-111    linear of order 2
ρ1611111-1-11-111-1-1-11-1-1-1-11-111-11-11-11    linear of order 2
ρ17222200-2-2-2-2-22202-20000000000000    orthogonal lifted from D4
ρ182222002-22-2-2-2-20220000000000000    orthogonal lifted from D4
ρ19222200-2-2-2-22-220-220000000000000    orthogonal lifted from D4
ρ202222002-22-222-20-2-20000000000000    orthogonal lifted from D4
ρ212-22-200020-2000-2i00-2i2i2i0000020-200    complex lifted from C4○D4
ρ222-22-200020-20002i002i-2i-2i0000020-200    complex lifted from C4○D4
ρ232-22-200020-2000-2i002i2i-2i00000-20200    complex lifted from C4○D4
ρ242-22-200020-20002i00-2i-2i2i00000-20200    complex lifted from C4○D4
ρ254-44-4000-4040000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4400-40400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-440040-400000000000000000000    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 C42.75C23
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 25 31 36)(2 26 32 33)(3 27 29 34)(4 28 30 35)(5 42 56 19)(6 43 53 20)(7 44 54 17)(8 41 55 18)(9 15 51 40)(10 16 52 37)(11 13 49 38)(12 14 50 39)(21 62 46 60)(22 63 47 57)(23 64 48 58)(24 61 45 59)
(1 61 31 59)(2 64 32 58)(3 63 29 57)(4 62 30 60)(5 52 56 10)(6 51 53 9)(7 50 54 12)(8 49 55 11)(13 18 38 41)(14 17 39 44)(15 20 40 43)(16 19 37 42)(21 35 46 28)(22 34 47 27)(23 33 48 26)(24 36 45 25)
(1 30 29 2)(3 32 31 4)(5 18 54 43)(6 42 55 17)(7 20 56 41)(8 44 53 19)(9 52 49 12)(10 11 50 51)(13 14 40 37)(15 16 38 39)(21 63 48 59)(22 58 45 62)(23 61 46 57)(24 60 47 64)(25 28 34 33)(26 36 35 27)
(1 9 3 11)(2 12 4 10)(5 64 7 62)(6 63 8 61)(13 25 15 27)(14 28 16 26)(17 21 19 23)(18 24 20 22)(29 49 31 51)(30 52 32 50)(33 39 35 37)(34 38 36 40)(41 45 43 47)(42 48 44 46)(53 57 55 59)(54 60 56 58)

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

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

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

Matrix representation of C42.75C23 in GL8(𝔽17)

001600000
00010000
10000000
016000000
000006160
0000110016
0000160011
000001660
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
601300000
060130000
1301100000
0130110000
000044512
00004131212
000012544
000055413
,
00100000
00010000
160000000
016000000
000001110
0000110016
0000160011
000001110
,
01000000
160000000
00010000
001600000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,6,0,0,16,0,0,0,0,16,0,0,6,0,0,0,0,0,16,11,0],[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],[6,0,13,0,0,0,0,0,0,6,0,13,0,0,0,0,13,0,11,0,0,0,0,0,0,13,0,11,0,0,0,0,0,0,0,0,4,4,12,5,0,0,0,0,4,13,5,5,0,0,0,0,5,12,4,4,0,0,0,0,12,12,4,13],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,11,16,0,0,0,0,0,11,0,0,1,0,0,0,0,1,0,0,11,0,0,0,0,0,16,11,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.75C23 in GAP, Magma, Sage, TeX

C_4^2._{75}C_2^3
% in TeX

G:=Group("C4^2.75C2^3");
// GroupNames label

G:=SmallGroup(128,2132);
// by ID

G=gap.SmallGroup(128,2132);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,436,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.75C23 in TeX

׿
×
𝔽