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G = C42.50C23  order 128 = 27

50th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.50C23, C4.602+ (1+4), C8⋊9D4⋊17C2, C4⋊C4.367D4, D4.Q8⋊36C2, C4⋊2Q16⋊39C2, D4⋊6D4.7C2, (C2×D4).171D4, C8.D4⋊25C2, C2.50(Q8○D8), C22⋊C4.50D4, D4.25(C4○D4), D4.7D4⋊45C2, C8.18D4⋊40C2, C4⋊C8.103C22, C4⋊C4.410C23, (C2×C8).189C23, (C2×C4).507C24, C23.325(C2×D4), C4⋊Q8.151C22, SD16⋊C4⋊36C2, C2.D8.60C22, C8⋊C4.44C22, C4.Q8.57C22, (C2×D4).424C23, (C4×D4).160C22, C22⋊C8.81C22, (C4×Q8).158C22, (C2×Q8).220C23, (C2×Q16).38C22, C2.143(D4⋊5D4), C22⋊Q8.82C22, C23.48D4⋊27C2, C23.20D4⋊35C2, C23.24D4⋊19C2, C23.36D4⋊18C2, (C22×C8).310C22, Q8⋊C4.71C22, (C2×SD16).57C22, C22.767(C22×D4), C42.C2.40C22, D4⋊C4.168C22, C2.86(D8⋊C22), (C22×C4).1151C23, C22.46C24⋊4C2, C42⋊C2.190C22, C42.30C22⋊10C2, (C2×M4(2)).114C22, C4.232(C2×C4○D4), (C2×C4).604(C2×D4), (C2×C4⋊C4).668C22, (C2×C4○D4).211C22, SmallGroup(128,2047)

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

Derived series
C1 — C2×C4 — C42.50C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — D4⋊6D4 — C42.50C23
Lower central
C1 — C2 — C2×C4 — C42.50C23
Upper central
C1 — C22 — C4×D4 — C42.50C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.50C23

Subgroups: 360 in 191 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×10], C8 [×4], C2×C4 [×5], C2×C4 [×19], D4 [×2], D4 [×6], Q8 [×6], C23 [×2], C23, C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×5], C4⋊C4 [×10], C2×C8 [×4], C2×C8, M4(2), SD16 [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C4○D4 [×6], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×7], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8 [×4], C22.D4 [×3], C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16 [×2], C2×C4○D4 [×2], C23.24D4, C23.36D4, C8⋊9D4, SD16⋊C4, D4.7D4 [×2], C4⋊2Q16, C8.18D4, C8.D4, D4.Q8, C23.48D4, C23.20D4, C42.30C22, D4⋊6D4, C22.46C24, C42.50C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ (1+4), D4⋊5D4, D8⋊C22, Q8○D8, C42.50C23

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

130000000
94000000
20400000
16013130000
00005057
0000120120
000012101210
00005500
,
160000000
016000000
001600000
000160000
00000100
000016000
0000116115
000010116
,
013100000
0130150000
161090000
00040000
000014300
00003300
0000314011
0000140140
,
160000000
016000000
09100000
04010000
00001000
000001600
00000010
000010116
,
134000000
94000000
216480000
16013130000
00005757
000050120
00005057
0000512107

G:=sub<GL(8,GF(17))| [13,9,2,16,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,12,12,5,0,0,0,0,0,0,10,5,0,0,0,0,5,12,12,0,0,0,0,0,7,0,10,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,0,16,0,0,0,0,0,13,13,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,9,4,0,0,0,0,0,0,0,0,14,3,3,14,0,0,0,0,3,3,14,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,11,0],[16,0,0,0,0,0,0,0,0,16,9,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16],[13,9,2,16,0,0,0,0,4,4,16,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,8,13,0,0,0,0,0,0,0,0,5,5,5,5,0,0,0,0,7,0,0,12,0,0,0,0,5,12,5,10,0,0,0,0,7,0,7,7] >;
 

Character table of C42.50C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114444222244444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111-1-1111-111-11-1-111-11-11-1    linear of order 2
ρ31111-111-111111111111-1-11-1-1-1-1-1-1-1    linear of order 2
ρ411111-1-1111-1-1111-111-1-11-1-1-11-11-11    linear of order 2
ρ51111-111-11111-11-111-1-1-1-1-1-1111111    linear of order 2
ρ611111-1-1111-1-1-11-1-11-11-111-11-11-11-1    linear of order 2
ρ7111111111111-11-111-1-111-11-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-1-111-1-1-11-1-11-111-111-11-11-11    linear of order 2
ρ91111-1-11-11111-1-1-1-1-11111-1-11111-1-1    linear of order 2
ρ10111111-1111-1-1-1-1-11-11-11-11-11-11-1-11    linear of order 2
ρ1111111-1111111-1-1-1-1-111-1-1-11-1-1-1-111    linear of order 2
ρ121111-11-1-111-1-1-1-1-11-11-1-1111-11-111-1    linear of order 2
ρ1311111-11111111-11-1-1-1-1-1-1111111-1-1    linear of order 2
ρ141111-11-1-111-1-11-111-1-11-11-111-11-1-11    linear of order 2
ρ151111-1-11-111111-11-1-1-1-1111-1-1-1-1-111    linear of order 2
ρ16111111-1111-1-11-111-1-111-1-1-1-11-111-1    linear of order 2
ρ1722220220-2-2-2-20-20-22000000000000    orthogonal lifted from D4
ρ1822220-2-20-2-2220-2022000000000000    orthogonal lifted from D4
ρ1922220-220-2-2-2-20202-2000000000000    orthogonal lifted from D4
ρ20222202-20-2-222020-2-2000000000000    orthogonal lifted from D4
ρ212-22-2-20022-2002i02i0000000002i02i00    complex lifted from C4○D4
ρ222-22-2200-22-2002i02i0000000002i02i00    complex lifted from C4○D4
ρ232-22-2-20022-2002i02i0000000002i02i00    complex lifted from C4○D4
ρ242-22-2200-22-2002i02i0000000002i02i00    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-400000000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-400000000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ294-4-440000004i4i00000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{50}C_2^3
 
% in TeX
 
G:=Group("C4^2.50C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2047);
 
// by ID
 
G=gap.SmallGroup(128,2047);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1*b^2,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*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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