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G = C42.55C23  order 128 = 27

55th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.55C23, C4.652+ (1+4), (D4×Q8)⋊8C2, C8⋊D4⋊41C2, C8⋊9D4⋊22C2, C8⋊8D4⋊53C2, C4⋊C4.160D4, Q8.Q8⋊37C2, Q8⋊D4⋊21C2, (C2×D4).320D4, C22⋊C4.53D4, D4.D4⋊22C2, C4⋊C4.238C23, C4⋊C8.106C22, (C2×C8).101C23, (C2×C4).512C24, Q8.26(C4○D4), C23.329(C2×D4), C4⋊Q8.153C22, SD16⋊C4⋊37C2, C8⋊C4.47C22, C2.78(D4○SD16), (C2×D4).238C23, (C4×D4).163C22, C4⋊D4.87C22, C22⋊C8.84C22, (C4×Q8).161C22, (C2×Q8).398C23, C2.148(D4⋊5D4), C2.D8.121C22, C4.Q8.106C22, C22⋊Q8.86C22, C23.38D4⋊14C2, C23.20D4⋊37C2, C23.48D4⋊28C2, (C22×C8).365C22, Q8⋊C4.73C22, (C2×SD16).58C22, C22.772(C22×D4), C22.8(C8.C22), C42.C2.42C22, D4⋊C4.141C22, (C22×C4).1156C23, (C22×Q8).345C22, C42.30C22⋊11C2, C42⋊C2.192C22, (C2×M4(2)).118C22, C22.47C24.1C2, C4.237(C2×C4○D4), (C2×C4).609(C2×D4), (C2×Q8⋊C4)⋊44C2, C2.77(C2×C8.C22), (C2×C4⋊C4).671C22, SmallGroup(128,2052)

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

Derived series
C1 — C2×C4 — C42.55C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — D4×Q8 — C42.55C23
Lower central
C1 — C2 — C2×C4 — C42.55C23
Upper central
C1 — C22 — C4×D4 — C42.55C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.55C23

Subgroups: 376 in 194 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×5], C2×C4 [×17], D4 [×7], Q8 [×2], Q8 [×9], C23 [×2], C23, C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×5], C4⋊C4 [×8], C2×C8 [×4], C2×C8, M4(2), SD16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×8], C8⋊C4, C22⋊C8 [×2], D4⋊C4, Q8⋊C4 [×9], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8 [×2], C22.D4, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×3], C22×Q8 [×2], C2×Q8⋊C4, C23.38D4, C8⋊9D4, SD16⋊C4, Q8⋊D4 [×2], D4.D4, C8⋊8D4, C8⋊D4, Q8.Q8, C23.48D4, C23.20D4, C42.30C22, D4×Q8, C22.47C24, C42.55C23

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○SD16, C42.55C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, ede-1=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 60 34 23)(2 57 35 24)(3 58 36 21)(4 59 33 22)(5 12 47 26)(6 9 48 27)(7 10 45 28)(8 11 46 25)(13 18 50 53)(14 19 51 54)(15 20 52 55)(16 17 49 56)(29 61 39 44)(30 62 40 41)(31 63 37 42)(32 64 38 43)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1300000
040000
004006
0004110
00011130
0060013
,
100000
010000
000100
0016000
000001
0000160
,
0130000
1300000
00011130
0011004
00130011
0004110
,
1600000
0160000
001341414
0044143
001414134
0014344
,
040000
1300000
00011130
0060013
004006
0004110

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,6,0,0,0,4,11,0,0,0,0,11,13,0,0,0,6,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,11,13,0,0,0,11,0,0,4,0,0,13,0,0,11,0,0,0,4,11,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,4,14,14,0,0,4,4,14,3,0,0,14,14,13,4,0,0,14,3,4,4],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,6,4,0,0,0,11,0,0,4,0,0,13,0,0,11,0,0,0,13,6,0] >;
 

Character table of C42.55C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11112248224444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111111-11111-111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ3111111111111-1111-11-1-111-1-1-1-1-1-1-1    linear of order 2
ρ41111111-11111111-11-111-1-11-1-1-1-1-1-1    linear of order 2
ρ5111111-1111-111-1-1-11-11-11-1-1-1-1-1-111    linear of order 2
ρ6111111-1-111-11-1-1-11-11-11-111-1-1-1-111    linear of order 2
ρ7111111-1111-11-1-1-1-1-1-1-111-111111-1-1    linear of order 2
ρ8111111-1-111-111-1-11111-1-11-11111-1-1    linear of order 2
ρ91111-1-111111-11-1-1-11-1-1-1-1111-11-1-11    linear of order 2
ρ101111-1-11-1111-1-1-1-11-11111-1-11-11-1-11    linear of order 2
ρ111111-1-111111-1-1-1-1-1-1-111-11-1-11-111-1    linear of order 2
ρ121111-1-11-1111-11-1-1111-1-11-11-11-111-1    linear of order 2
ρ131111-1-1-1111-1-1111111-11-1-1-1-11-11-11    linear of order 2
ρ141111-1-1-1-111-1-1-111-1-1-11-1111-11-11-11    linear of order 2
ρ151111-1-1-1111-1-1-1111-111-1-1-111-11-11-1    linear of order 2
ρ161111-1-1-1-111-1-1111-11-1-1111-11-11-11-1    linear of order 2
ρ172222-2-2-20-2-2220-2200000000000000    orthogonal lifted from D4
ρ182222-2-220-2-2-2202-200000000000000    orthogonal lifted from D4
ρ1922222220-2-2-2-20-2200000000000000    orthogonal lifted from D4
ρ20222222-20-2-22-202-200000000000000    orthogonal lifted from D4
ρ212-22-20000-2200-2002i22i0000002i02i00    complex lifted from C4○D4
ρ222-22-20000-22002002i-22i0000002i02i00    complex lifted from C4○D4
ρ232-22-20000-22002002i-22i0000002i02i00    complex lifted from C4○D4
ρ242-22-20000-2200-2002i22i0000002i02i00    complex lifted from C4○D4
ρ254-44-400004-40000000000000000000    orthogonal lifted from 2+ (1+4)
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._{55}C_2^3
 
% in TeX
 
G:=Group("C4^2.55C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2052);
 
// by ID
 
G=gap.SmallGroup(128,2052);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,352,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,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^-1=a^2*b^2*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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