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G = C42.58C23  order 128 = 27

58th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.58C23, C4.792- (1+4), C8⋊Q8⋊24C2, C4⋊C4.166D4, C8⋊9D4.2C2, Q8.Q8⋊40C2, Q8⋊Q8⋊21C2, (C2×D4).330D4, C8.32(C4○D4), C8.D4⋊31C2, D4⋊3Q8.6C2, Q16⋊C4⋊25C2, C8.18D4⋊31C2, C4⋊C8.117C22, C4⋊C4.249C23, (C2×C8).364C23, (C2×C4).536C24, C22⋊C4.176D4, C23.481(C2×D4), C4⋊Q8.168C22, C2.89(D4⋊6D4), C4.Q8.66C22, C8⋊C4.50C22, C2.89(D4○SD16), (C4×D4).176C22, C22⋊C8.95C22, (C4×Q8).177C22, (C2×Q16).88C22, (C2×Q8).240C23, M4(2)⋊C4⋊31C2, C2.D8.129C22, C23.20D4⋊43C2, C23.47D4⋊21C2, C23.48D4⋊32C2, (C22×C4).340C23, (C22×C8).287C22, Q8⋊C4.77C22, C22.796(C22×D4), C22.6(C8.C22), C22⋊Q8.103C22, C42.C2.49C22, C42⋊C2.207C22, (C2×M4(2)).129C22, C22.46C24.3C2, (C2×C4.Q8)⋊12C2, C4.118(C2×C4○D4), (C2×C4).620(C2×D4), C2.82(C2×C8.C22), (C2×C4⋊C4).685C22, SmallGroup(128,2076)

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

Derived series
C1 — C2×C4 — C42.58C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.46C24 — C42.58C23
Lower central
C1 — C2 — C2×C4 — C42.58C23
Upper central
C1 — C22 — C4×D4 — C42.58C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.58C23

Subgroups: 296 in 172 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×5], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×7], C4⋊C4 [×12], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8, C8⋊C4, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×6], C2.D8 [×3], C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4, C42.C2, C42.C2 [×2], C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×Q16, C2×C4.Q8, M4(2)⋊C4, C8⋊9D4, Q16⋊C4, C8.18D4, C8.D4, Q8⋊Q8, Q8.Q8, C23.47D4 [×2], C23.48D4, C23.20D4, C8⋊Q8, C22.46C24, D4⋊3Q8, C42.58C23

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⋊6D4, C2×C8.C22, D4○SD16, C42.58C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=a2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1150000
1160000
0016000
0001600
000010
000001
,
100000
010000
000100
0016000
0000016
000010
,
1380000
040000
00101600
0016700
0000110
00001016
,
400000
040000
000010
000001
001000
000100
,
100000
1160000
001000
000100
0000160
0000016

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,1,10,0,0,0,0,10,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
 

Character table of C42.58C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11112242244444444888888444488
ρ111111111111111111111111111111    trivial
ρ2111111111111-1111-1-1-1111-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11111-1-1-11-11-1-11-11-111-11-1-11    linear of order 2
ρ41111-1-11111-1-111-1111-1-11-1-1-11-111-1    linear of order 2
ρ5111111-1111-11-1-1-11-111-1-11-1-1-1-1-111    linear of order 2
ρ6111111-1111-111-1-111-1-1-1-1111111-1-1    linear of order 2
ρ71111-1-1-11111-11-1111-111-1-1-1-11-11-11    linear of order 2
ρ81111-1-1-11111-1-1-111-11-11-1-111-11-11-1    linear of order 2
ρ9111111-111-1-11-1-1-1-1-11111-1-11111-1-1    linear of order 2
ρ10111111-111-1-111-1-1-11-1-111-11-1-1-1-111    linear of order 2
ρ111111-1-1-111-11-11-11-11-11-111-11-11-11-1    linear of order 2
ρ121111-1-1-111-11-1-1-11-1-11-1-1111-11-11-11    linear of order 2
ρ13111111111-111111-1111-1-1-11-1-1-1-1-1-1    linear of order 2
ρ14111111111-111-111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ151111-1-1111-1-1-1-11-1-1-1-111-111-11-111-1    linear of order 2
ρ161111-1-1111-1-1-111-1-111-11-11-11-11-1-11    linear of order 2
ρ17222222-2-2-20-2-202200000000000000    orthogonal lifted from D4
ρ182222222-2-202-20-2-200000000000000    orthogonal lifted from D4
ρ192222-2-22-2-20-220-2200000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-202202-200000000000000    orthogonal lifted from D4
ρ212-22-20002-22i002i002i2i0000000-20200    complex lifted from C4○D4
ρ222-22-20002-22i002i002i2i000000020-200    complex lifted from C4○D4
ρ232-22-20002-22i002i002i2i000000020-200    complex lifted from C4○D4
ρ242-22-20002-22i002i002i2i0000000-20200    complex lifted from C4○D4
ρ254-44-4000-4400000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{58}C_2^3
 
% in TeX
 
G:=Group("C4^2.58C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2076);
 
// by ID
 
G=gap.SmallGroup(128,2076);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,100,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=e*a*e=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,e*c*e=a^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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