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G = C42.49C23  order 128 = 27

49th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.49C23, C4.592+ (1+4), C8⋊9D4⋊16C2, C8⋊D4⋊38C2, C4⋊C8⋊35C22, C4⋊C4.157D4, D4⋊3Q8⋊3C2, C4⋊Q8⋊24C22, D4⋊Q8⋊34C2, D4⋊5D4.4C2, (C2×D4).317D4, C2.48(D4○D8), (C2×C8).98C23, (C4×Q8)⋊28C22, (C2×Q16)⋊9C22, C2.D8⋊37C22, C8⋊C4⋊24C22, D4.24(C4○D4), C22⋊SD16⋊22C2, D4.7D4⋊44C2, C8.18D4⋊39C2, C4⋊C4.235C23, C22⋊C8⋊31C22, (C2×C4).506C24, Q8.D4⋊42C2, C22⋊C4.167D4, C23.475(C2×D4), C22⋊Q8⋊19C22, SD16⋊C4⋊35C2, Q8⋊C4⋊44C22, (C2×D4).423C23, (C4×D4).159C22, C4⋊D4.84C22, C22.D8⋊28C2, (C2×Q8).219C23, C2.142(D4⋊5D4), D4⋊C4.72C22, C23.36D4⋊17C2, C23.48D4⋊26C2, (C2×M4(2))⋊28C22, (C22×C8).309C22, (C2×SD16).56C22, C4.4D4.66C22, C22.766(C22×D4), C22.5(C8.C22), (C22×C4).1150C23, (C22×D4).412C22, C42.28C22⋊16C2, (C2×C4⋊C4)⋊59C22, C4.231(C2×C4○D4), (C2×C4).603(C2×D4), (C2×D4⋊C4)⋊31C2, C2.75(C2×C8.C22), (C2×C4○D4).210C22, SmallGroup(128,2046)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.49C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — D4⋊5D4 — C42.49C23
Lower central
C1 — C2 — C2×C4 — C42.49C23
Upper central
C1 — C22 — C4×D4 — C42.49C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.49C23

Subgroups: 448 in 205 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×16], C8 [×4], C2×C4 [×5], C2×C4 [×15], D4 [×2], D4 [×10], Q8 [×5], C23 [×2], C23 [×7], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×5], C4⋊C4 [×7], C2×C8 [×4], C2×C8, M4(2), SD16 [×3], Q16, C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4 [×3], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×4], C4⋊C8, C2.D8 [×3], C2×C22⋊C4, C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16, C22×D4, C2×C4○D4, C2×D4⋊C4, C23.36D4, C8⋊9D4, SD16⋊C4, C22⋊SD16, D4.7D4, Q8.D4, C8.18D4, C8⋊D4, D4⋊Q8, C22.D8, C23.48D4, C42.28C22, D4⋊5D4, D4⋊3Q8, C42.49C23

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), D4⋊5D4, C2×C8.C22, D4○D8, C42.49C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2b2, ab=ba, cac-1=eae=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >

Smallest permutation representation
►On 32 points
Generators in S32
(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)

(1 19 25 21)(2 20 26 22)(3 17 27 23)(4 18 28 24)(5 15 9 32)(6 16 10 29)(7 13 11 30)(8 14 12 31)

(1 6 27 12)(2 5 28 11)(3 8 25 10)(4 7 26 9)(13 20 32 24)(14 19 29 23)(15 18 30 22)(16 17 31 21)

(1 27)(2 4)(3 25)(5 30)(6 14)(7 32)(8 16)(9 13)(10 31)(11 15)(12 29)(17 19)(18 22)(20 24)(21 23)(26 28)

(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)
 
G:=sub<Sym(32)| (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), (1,19,25,21)(2,20,26,22)(3,17,27,23)(4,18,28,24)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,6,27,12)(2,5,28,11)(3,8,25,10)(4,7,26,9)(13,20,32,24)(14,19,29,23)(15,18,30,22)(16,17,31,21), (1,27)(2,4)(3,25)(5,30)(6,14)(7,32)(8,16)(9,13)(10,31)(11,15)(12,29)(17,19)(18,22)(20,24)(21,23)(26,28), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)>;
 
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), (1,19,25,21)(2,20,26,22)(3,17,27,23)(4,18,28,24)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,6,27,12)(2,5,28,11)(3,8,25,10)(4,7,26,9)(13,20,32,24)(14,19,29,23)(15,18,30,22)(16,17,31,21), (1,27)(2,4)(3,25)(5,30)(6,14)(7,32)(8,16)(9,13)(10,31)(11,15)(12,29)(17,19)(18,22)(20,24)(21,23)(26,28), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31) );
 
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)], [(1,19,25,21),(2,20,26,22),(3,17,27,23),(4,18,28,24),(5,15,9,32),(6,16,10,29),(7,13,11,30),(8,14,12,31)], [(1,6,27,12),(2,5,28,11),(3,8,25,10),(4,7,26,9),(13,20,32,24),(14,19,29,23),(15,18,30,22),(16,17,31,21)], [(1,27),(2,4),(3,25),(5,30),(6,14),(7,32),(8,16),(9,13),(10,31),(11,15),(12,29),(17,19),(18,22),(20,24),(21,23),(26,28)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31)])
 

Matrix representation ►G ⊆ GL6(𝔽17)

120000
16160000
000010
00161115
0016000
00161016
,
100000
010000
000100
0016000
00161115
00160116
,
1300000
440000
0014300
003300
0031406
003030
,
100000
010000
001000
0001600
0000160
00016161
,
120000
0160000
000010
00161115
001000
0000016

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,16,16,16,0,0,0,1,0,1,0,0,1,1,0,0,0,0,0,15,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,16,0,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,15,16],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,14,3,3,3,0,0,3,3,14,0,0,0,0,0,0,3,0,0,0,0,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,1],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,15,0,16] >;
 

Character table of C42.49C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11112244482244444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-1-1111-11-1-11-111-1-11-111-1-11    linear of order 2
ρ3111111-1-1-1-1111-11-1-11-11-111-1-1-1-111    linear of order 2
ρ41111-1-1111-111-1-1-11-1-1-111-111-1-11-11    linear of order 2
ρ5111111111111-11111-1-11-11-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-1-1-1-111111-1-111-111-1-11-1-111-1    linear of order 2
ρ7111111-1-1-1-111-1-11-1-1-11111-11111-1-1    linear of order 2
ρ81111-1-1111-1111-1-11-1111-1-1-1-111-11-1    linear of order 2
ρ911111111-1111-1-11-1-1-11-11-1-1-1-1-1-111    linear of order 2
ρ101111-1-1-1-111111-1-11-111-1-11-11-1-11-11    linear of order 2
ρ11111111-1-11-111-11111-1-1-1-1-1-1111111    linear of order 2
ρ121111-1-111-1-11111-1-111-1-111-1-111-1-11    linear of order 2
ρ1311111111-11111-11-1-11-1-1-1-111111-1-1    linear of order 2
ρ141111-1-1-1-11111-1-1-11-1-1-1-1111-111-11-1    linear of order 2
ρ15111111-1-11-1111111111-11-11-1-1-1-1-1-1    linear of order 2
ρ161111-1-111-1-111-11-1-11-11-1-1111-1-111-1    linear of order 2
ρ172222220020-2-202-2-2-2000000000000    orthogonal lifted from D4
ρ182222-2-200-20-2-20222-2000000000000    orthogonal lifted from D4
ρ1922222200-20-2-20-2-222000000000000    orthogonal lifted from D4
ρ202222-2-20020-2-20-22-22000000000000    orthogonal lifted from D4
ρ212-22-2002-200-222i00002i0000002i2i000    complex lifted from C4○D4
ρ222-22-200-2200-222i00002i0000002i2i000    complex lifted from C4○D4
ρ232-22-2002-200-222i00002i0000002i2i000    complex lifted from C4○D4
ρ242-22-200-2200-222i00002i0000002i2i000    complex lifted from C4○D4
ρ2544-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ264-44-40000004-400000000000000000    orthogonal lifted from 2+ (1+4)
ρ2744-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ284-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ294-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{49}C_2^3
 
% in TeX
 
G:=Group("C4^2.49C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2046);
 
// by ID
 
G=gap.SmallGroup(128,2046);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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