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G = C42.479C23  order 128 = 27

340th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.479C23, C4.752+ (1+4), (C4×D8)⋊44C2, D4⋊7(C4○D4), C8⋊D4⋊48C2, C8⋊6D4⋊17C2, C4⋊C4.375D4, D4⋊6D4⋊12C2, D4⋊D4⋊49C2, D4⋊Q8⋊37C2, (C2×D4).325D4, C22⋊C4.58D4, C2.53(Q8○D8), D4.D4⋊24C2, C4⋊C4.422C23, C4⋊C8.112C22, C4.48(C8⋊C22), (C2×C8).193C23, (C4×C8).228C22, (C2×C4).522C24, C4.SD16⋊21C2, C23.339(C2×D4), C4⋊Q8.157C22, (C4×D4).171C22, (C2×D4).245C23, (C2×D8).141C22, C4⋊D4.94C22, C22⋊C8.90C22, (C2×Q8).230C23, C2.158(D4⋊5D4), C2.D8.126C22, C22⋊Q8.93C22, D4⋊C4.76C22, C23.48D4⋊30C2, C23.36D4⋊25C2, (C22×C4).335C23, Q8⋊C4.16C22, (C2×SD16).61C22, C22.782(C22×D4), (C2×M4(2)).124C22, C4.247(C2×C4○D4), (C2×C4).615(C2×D4), C2.80(C2×C8⋊C22), (C2×C4⋊C4).674C22, (C2×C4○D4).220C22, SmallGroup(128,2062)

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

Derived series
C1 — C2×C4 — C42.479C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — D4⋊6D4 — C42.479C23
Lower central
C1 — C2 — C2×C4 — C42.479C23
Upper central
C1 — C22 — C4×D4 — C42.479C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.479C23

Subgroups: 432 in 210 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×13], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×2], D4 [×13], Q8 [×6], C23 [×2], C23 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×6], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C2×Q8, C4○D4 [×10], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8, C2.D8, C2.D8 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×3], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C22.D4 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C2×C4○D4 [×2], C2×C4○D4, C23.36D4 [×2], C8⋊6D4, C4×D8, D4⋊D4 [×2], D4.D4, C8⋊D4 [×2], D4⋊Q8, C23.48D4 [×2], C4.SD16, D4⋊6D4 [×2], C42.479C23

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, Q8○D8, C42.479C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2b2, ab=ba, cac-1=a-1, dad=ab2, eae=a-1b2, 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 20 39 23)(2 17 40 24)(3 18 37 21)(4 19 38 22)(5 30 61 28)(6 31 62 25)(7 32 63 26)(8 29 64 27)(9 54 15 59)(10 55 16 60)(11 56 13 57)(12 53 14 58)(33 43 49 46)(34 44 50 47)(35 41 51 48)(36 42 52 45)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1300000
1640000
0000013
000040
0001300
004000
,
100000
010000
000100
0016000
000001
0000160
,
420000
0130000
0001300
0013000
000004
000040
,
100000
010000
0014300
003300
0000143
000033
,
13150000
1640000
000400
0013000
0000013
000040

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

Character table of C42.479C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114444822224444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-1-11111-1-11-1-111-1111111-1-1    linear of order 2
ρ31111-1-1-11-1-111-11-1-1111-11-111-11-11-1    linear of order 2
ρ41111111-11-111-1-11-1-1-11-1-1-111-11-1-11    linear of order 2
ρ51111-1-1-111-111-1-1-1-11-1-1-1111-11-11-11    linear of order 2
ρ61111111-1-1-111-111-1-11-1-1-111-11-111-1    linear of order 2
ρ711111111-11111-1111-1-111-11-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-1-1111111-11-11-11-1-11-1-1-1-111    linear of order 2
ρ91111-11-11-11111-1111-1-1-1-1-1-1111111    linear of order 2
ρ1011111-11-1111111-11-11-1-11-1-11111-1-1    linear of order 2
ρ1111111-1111-111-1-1-1-11-1-11-11-11-11-11-1    linear of order 2
ρ121111-11-1-1-1-111-111-1-11-1111-11-11-1-11    linear of order 2
ρ1311111-111-1-111-11-1-11111-1-1-1-11-11-11    linear of order 2
ρ141111-11-1-11-111-1-11-1-1-1111-1-1-11-111-1    linear of order 2
ρ151111-11-1111111111111-1-11-1-1-1-1-1-1-1    linear of order 2
ρ1611111-11-1-11111-1-11-1-11-111-1-1-1-1-111    linear of order 2
ρ172222020202-2-220-2-2-2000000000000    orthogonal lifted from D4
ρ1822220-20-202-2-2202-22000000000000    orthogonal lifted from D4
ρ1922220-2020-2-2-2-2022-2000000000000    orthogonal lifted from D4
ρ202222020-20-2-2-2-20-222000000000000    orthogonal lifted from D4
ρ212-22-2-2020002-202i0002i0000002i02i00    complex lifted from C4○D4
ρ222-22-220-20002-202i0002i0000002i02i00    complex lifted from C4○D4
ρ232-22-220-20002-202i0002i0000002i02i00    complex lifted from C4○D4
ρ242-22-2-2020002-202i0002i0000002i02i00    complex lifted from C4○D4
ρ254-4-4400000-40040000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-4000000-4400000000000000000    orthogonal lifted from 2+ (1+4)
ρ274-4-4400000400-40000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-400000000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{479}C_2^3
 
% in TeX
 
G:=Group("C4^2.479C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2062);
 
// by ID
 
G=gap.SmallGroup(128,2062);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,2019,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,e*a*e=a^-1*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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