Copied to
clipboard

G = C42.25C23order 128 = 27

25th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.25C23, C4.142- 1+4, C4.332+ 1+4, C4⋊C4.140D4, C88D4.2C2, C8⋊D4.3C2, D4.Q831C2, Q8.Q831C2, Q8⋊Q815C2, C4.Q1632C2, C42Q1632C2, C8.D417C2, C4⋊C8.86C22, C2.34(Q8○D8), C22⋊C4.32D4, C23.93(C2×D4), D4.D415C2, C8.18D420C2, C4⋊C4.197C23, (C2×C8).340C23, (C2×C4).456C24, Q8.D431C2, C4⋊Q8.128C22, C2.D8.49C22, C4.Q8.96C22, C2.53(D4○SD16), (C2×D4).197C23, (C4×D4).135C22, D4⋊C4.8C22, C4⋊D4.51C22, (C2×Q16).77C22, (C2×Q8).185C23, (C4×Q8).132C22, C22⋊Q8.51C22, (C22×C8).191C22, Q8⋊C4.60C22, (C2×SD16).91C22, C4.4D4.46C22, C22.716(C22×D4), C42.C2.31C22, C22.35C247C2, C42.6C2213C2, (C22×C4).1111C23, (C2×M4(2)).94C22, C42⋊C2.174C22, C22.36C24.5C2, C2.75(C22.31C24), (C2×C4).580(C2×D4), SmallGroup(128,1990)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.25C23
C1C2C4C2×C4C42C4×D4C22.36C24 — C42.25C23
C1C2C2×C4 — C42.25C23
C1C22C42⋊C2 — C42.25C23
C1C2C2C2×C4 — C42.25C23

Generators and relations for C42.25C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b2, ab=ba, cac-1=a-1, dad=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd=a2c, ece=bc, ede=a2d >

Subgroups: 308 in 167 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×3], Q8 [×7], C23, C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8, M4(2), SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×3], C2×Q8, D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×3], C4⋊D4, C22⋊Q8 [×3], C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2 [×2], C422C2 [×3], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C42.6C22, D4.D4, C42Q16, Q8.D4 [×2], C88D4, C8.18D4, C8⋊D4, C8.D4, Q8⋊Q8, C4.Q16, D4.Q8, Q8.Q8, C22.35C24, C22.36C24, C42.25C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○SD16, Q8○D8, C42.25C23

Character table of C42.25C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114822444448888888444488
ρ111111111111111111111111111    trivial
ρ21111-11111-1-11-1-11-1-11-111-11-1-11    linear of order 2
ρ31111-1111-11-1-111-1-111-1-1-11-11-11    linear of order 2
ρ411111111-1-11-1-1-1-11-111-1-1-1-1-111    linear of order 2
ρ51111-11111-1-11-1-1-111-1-11-11-111-1    linear of order 2
ρ611111111111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ711111111-1-11-1-1-11-11-11-11111-1-1    linear of order 2
ρ81111-1111-11-1-11111-1-1-1-11-11-11-1    linear of order 2
ρ91111-1-1111-1-11-11-111-11-11-11-1-11    linear of order 2
ρ1011111-11111111-1-1-1-1-1-1-1111111    linear of order 2
ρ1111111-111-1-11-1-111-11-1-11-1-1-1-111    linear of order 2
ρ121111-1-111-11-1-11-111-1-111-11-11-11    linear of order 2
ρ1311111-11111111-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-1-1111-1-11-111-1-111-1-11-111-1    linear of order 2
ρ151111-1-111-11-1-11-1-1-111111-11-11-1    linear of order 2
ρ1611111-111-1-11-1-11-11-11-111111-1-1    linear of order 2
ρ17222220-2-22-2-2-220000000000000    orthogonal lifted from D4
ρ182222-20-2-2-2-22220000000000000    orthogonal lifted from D4
ρ19222220-2-2-22-22-20000000000000    orthogonal lifted from D4
ρ202222-20-2-2222-2-20000000000000    orthogonal lifted from D4
ρ214-44-4004-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2344-4-400000000000000000220-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-400000000000000000-2202200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-440000000000000000-2-202-2000    complex lifted from D4○SD16
ρ264-4-4400000000000000002-20-2-2000    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C42.25C23 in GL8(𝔽17)

115000000
116000000
016010000
1161600000
00000010
000016161615
00001000
00000001
,
160000000
016000000
001600000
000160000
00000100
000016000
000016161615
00001011
,
007100000
50700000
051250000
1251250000
000012500
00005500
0000121277
000050510
,
101500000
001610000
001600000
011600000
00006010
00001651615
0000160110
00001212012
,
10000000
01000000
101600000
100160000
00001000
000001600
00000010
00001601616

G:=sub<GL(8,GF(17))| [1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,5,0,12,0,0,0,0,0,0,5,5,0,0,0,0,7,7,12,12,0,0,0,0,10,0,5,5,0,0,0,0,0,0,0,0,12,5,12,5,0,0,0,0,5,5,12,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,7,10],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,16,16,12,0,0,0,0,0,5,0,12,0,0,0,0,1,16,11,0,0,0,0,0,0,15,0,12],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16] >;

C42.25C23 in GAP, Magma, Sage, TeX

C_4^2._{25}C_2^3
% in TeX

G:=Group("C4^2.25C2^3");
// GroupNames label

G:=SmallGroup(128,1990);
// by ID

G=gap.SmallGroup(128,1990);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,675,1018,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.25C23 in TeX

׿
×
𝔽