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G = C42.73C23  order 128 = 27

73rd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.73C23, C4.1492+ (1+4), C4⋊C4.185D4, C8⋊4Q8⋊20C2, Q8⋊3Q8⋊13C2, C4⋊2Q16⋊44C2, C4⋊Q16⋊24C2, C8.20(C4○D4), (C2×Q8).253D4, C2.72(Q8○D8), (C4×SD16).8C2, C8.2D4.3C2, Q16⋊C4⋊30C2, C4⋊C4.449C23, C4⋊C8.151C22, (C2×C8).222C23, (C2×C4).590C24, (C4×C8).208C22, D4.D4.2C2, C4⋊Q8.217C22, Q8.D4.4C2, C8⋊C4.77C22, C2.44(Q8⋊6D4), (C2×D4).284C23, (C4×D4).223C22, C4.54(C8.C22), (C2×Q8).269C23, (C4×Q8).213C22, (C2×Q16).94C22, C4.Q8.141C22, Q8⋊C4.95C22, (C2×SD16).76C22, C4.4D4.90C22, C22.850(C22×D4), D4⋊C4.177C22, C22.50C24.11C2, C4.168(C2×C4○D4), (C2×C4).654(C2×D4), C2.92(C2×C8.C22), SmallGroup(128,2130)

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

Derived series
C1 — C2×C4 — C42.73C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.50C24 — C42.73C23
Lower central
C1 — C2 — C2×C4 — C42.73C23
Upper central
C1 — C22 — C4×Q8 — C42.73C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.73C23

Subgroups: 304 in 175 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×12], C22, C22 [×3], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×2], Q8 [×12], C23, C42, C42 [×2], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×13], C2×C8 [×2], C2×C8 [×2], SD16 [×4], Q16 [×8], C22×C4, C2×D4, C2×Q8 [×2], C2×Q8 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C4.Q8, C42⋊C2, C4×D4, C4×Q8 [×2], C4×Q8 [×4], C4×Q8, C22⋊Q8, C4.4D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×SD16, C2×SD16 [×2], C2×Q16 [×6], C4×SD16, Q16⋊C4 [×2], C8⋊4Q8, D4.D4, C4⋊2Q16, C4⋊2Q16 [×2], Q8.D4 [×2], C4⋊Q16, C8.2D4 [×2], C22.50C24, Q8⋊3Q8, C42.73C23

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), Q8⋊6D4, C2×C8.C22, Q8○D8, C42.73C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=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 >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

00100000
00010000
160000000
016000000
000012050
000001205
00005050
00000505
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
0014140000
001430000
1414000000
143000000
000000314
0000001414
000014300
00003300
,
016000000
10000000
000160000
00100000
000000160
00000001
00001000
000001600
,
01000000
160000000
000160000
00100000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,5,0,0,0,0,0,0,12,0,5,0,0,0,0,5,0,5,0,0,0,0,0,0,5,0,5],[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],[0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

Character table of C42.73C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R8A8B8C8D8E8F
 size 11118222244444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111111-1-1-1-11-1-1-11-1-11-11-11-111-1-11    linear of order 2
ρ3111111111-1-1-111-1-1-11-111-1-1-1-1-1-111    linear of order 2
ρ41111111-1-111-1-1-11-11-1-1-111-11-1-11-11    linear of order 2
ρ51111-111111-1-11-1-1-11-11-1-111-1-1-1-111    linear of order 2
ρ61111-111-1-1-11-1-111-1-1111-1-111-1-11-11    linear of order 2
ρ71111-11111-1111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111-111-1-11-11-11-1111-11-11-1-111-1-11    linear of order 2
ρ91111111-1-1-11-1-1-11-1-1-1-11-111-111-11-1    linear of order 2
ρ101111111111-1-111-1-111-1-1-1-111111-1-1    linear of order 2
ρ111111111-1-11-11-1-1-111-111-1-1-11-1-111-1    linear of order 2
ρ12111111111-1111111-111-1-11-1-1-1-1-1-1-1    linear of order 2
ρ131111-111-1-1-1-11-11-11-11-1-11111-1-111-1    linear of order 2
ρ141111-111111111-1111-1-111-11-1-1-1-1-1-1    linear of order 2
ρ151111-111-1-111-1-111-1111-11-1-1-111-11-1    linear of order 2
ρ161111-11111-1-1-11-1-1-1-1-11111-11111-1-1    linear of order 2
ρ1722220-2-222022-20-2-20000000000000    orthogonal lifted from D4
ρ1822220-2-2-2-20-22202-20000000000000    orthogonal lifted from D4
ρ1922220-2-2-2-202-220-220000000000000    orthogonal lifted from D4
ρ2022220-2-2220-2-2-20220000000000000    orthogonal lifted from D4
ρ212-22-202-2002i0002i002i2i000000-22000    complex lifted from C4○D4
ρ222-22-202-2002i0002i002i2i000000-22000    complex lifted from C4○D4
ρ232-22-202-2002i0002i002i2i0000002-2000    complex lifted from C4○D4
ρ242-22-202-2002i0002i002i2i0000002-2000    complex lifted from C4○D4
ρ254-44-40-440000000000000000000000    orthogonal lifted from 2+ (1+4)
ρ264-4-440004-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44000-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000002√2002√200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-400000000000000000002√2002√200    symplectic lifted from Q8○D8, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{73}C_2^3
 
% in TeX
 
G:=Group("C4^2.73C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2130);
 
// by ID
 
G=gap.SmallGroup(128,2130);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,100,346,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^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
 

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