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G = C42.477C23  order 128 = 27

338th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.477C23, C4.732+ (1+4), (D4×Q8).7C2, C4⋊C4.373D4, (C4×Q16)⋊43C2, C8⋊6D4.5C2, Q8⋊Q8⋊19C2, C4⋊2Q16⋊40C2, (C2×D4).323D4, C8.D4⋊29C2, C22⋊C4.56D4, C2.52(Q8○D8), C4⋊C4.420C23, C4⋊C8.110C22, (C4×C8).227C22, (C2×C4).520C24, (C2×C8).192C23, Q8.30(C4○D4), C22⋊Q16⋊34C2, C4.SD16⋊20C2, C23.337(C2×D4), C4⋊Q8.155C22, C4.Q8.62C22, (C4×D4).169C22, C4.77(C8.C22), C22⋊C8.88C22, (C2×Q8).229C23, (C4×Q8).165C22, C2.156(D4⋊5D4), C2.D8.193C22, C22⋊Q8.91C22, C23.38D4⋊18C2, C23.20D4⋊41C2, (C22×C4).333C23, (C2×Q16).136C22, Q8⋊C4.15C22, C22.780(C22×D4), (C22×Q8).349C22, C42⋊C2.198C22, (C2×M4(2)).122C22, C22.50C24.4C2, C4.245(C2×C4○D4), (C2×C4).613(C2×D4), C2.78(C2×C8.C22), SmallGroup(128,2060)

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

Derived series
C1 — C2×C4 — C42.477C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — D4×Q8 — C42.477C23
Lower central
C1 — C2 — C2×C4 — C42.477C23
Upper central
C1 — C22 — C4×D4 — C42.477C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.477C23

Subgroups: 336 in 187 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×2], Q8 [×2], Q8 [×13], C23 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], Q16 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×2], Q8⋊C4 [×8], C4⋊C8, C4.Q8 [×2], C2.D8, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×4], C22⋊Q8 [×2], C4.4D4, C42⋊2C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×M4(2) [×2], C2×Q16, C2×Q16 [×2], C22×Q8 [×2], C23.38D4 [×2], C8⋊6D4, C4×Q16, C22⋊Q16 [×2], C4⋊2Q16, C8.D4 [×2], Q8⋊Q8, C23.20D4 [×2], C4.SD16, D4×Q8, C22.50C24, C42.477C23

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, C42.477C23

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

01000000
160000000
00010000
001600000
000000016
00000010
00000100
000016000
,
160000000
016000000
001600000
000160000
000001600
00001000
000000016
00000010
,
012030000
120300000
014050000
140500000
00004000
000001300
000000130
00000004
,
00100000
00010000
10000000
01000000
000051200
0000121200
000000512
0000001212
,
160000000
01000000
001600000
00010000
00000100
000016000
000000016
00000010

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

Character table of C42.477C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11114422224444444888888444488
ρ111111111111111111111111111111    trivial
ρ211111-1-111-11-1-11-1-111-1-11-11-111-11-1    linear of order 2
ρ31111-1-1111111-1-1111-11-1-1-111111-1-1    linear of order 2
ρ41111-11-111-11-11-1-1-11-1-11-111-111-1-11    linear of order 2
ρ51111111111-1-111-11-1-1-1-1-1-1-1111111    linear of order 2
ρ611111-1-111-1-11-111-1-1-111-11-1-111-11-1    linear of order 2
ρ71111-1-11111-1-1-1-1-11-11-1111-11111-1-1    linear of order 2
ρ81111-11-111-1-111-11-1-111-11-1-1-111-1-11    linear of order 2
ρ91111-1-11111-11-1-111-11-11-1-11-1-1-1-111    linear of order 2
ρ101111-11-111-1-1-11-1-1-1-111-1-1111-1-111-1    linear of order 2
ρ111111111111-111111-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ1211111-1-111-1-1-1-11-1-1-1-1111-111-1-11-11    linear of order 2
ρ131111-1-111111-1-1-1-111-11-111-1-1-1-1-111    linear of order 2
ρ141111-11-111-1111-11-11-1-111-1-11-1-111-1    linear of order 2
ρ1511111111111-111-111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ1611111-1-111-111-111-111-1-1-11-11-1-11-11    linear of order 2
ρ172222-22-2-2-2-200-22020000000000000    orthogonal lifted from D4
ρ182222222-2-2200-2-20-20000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-2002-2020000000000000    orthogonal lifted from D4
ρ202222-2-22-2-2200220-20000000000000    orthogonal lifted from D4
ρ212-22-2000-2202i-200202i0000002i002i00    complex lifted from C4○D4
ρ222-22-2000-2202i200-202i0000002i002i00    complex lifted from C4○D4
ρ232-22-2000-2202i-200202i0000002i002i00    complex lifted from C4○D4
ρ242-22-2000-2202i200-202i0000002i002i00    complex lifted from C4○D4
ρ254-44-40004-400000000000000000000    orthogonal lifted from 2+ (1+4)
ρ264-4-4400400-40000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-4400-40040000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-4000000000000000000002√22√2000    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-4000000000000000000002√22√2000    symplectic lifted from Q8○D8, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{477}C_2^3
 
% in TeX
 
G:=Group("C4^2.477C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2060);
 
// by ID
 
G=gap.SmallGroup(128,2060);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,352,2019,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,e*a*e^-1=a^-1*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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