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G = C42.517C23  order 128 = 27

378th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.517C23, C4.382- (1+4), C4⋊C4.182D4, C8⋊4Q8⋊12C2, D4.Q8⋊50C2, Q8.Q8⋊49C2, Q8⋊Q8⋊27C2, D4⋊3Q8⋊14C2, (C4×SD16)⋊64C2, D8⋊C4.1C2, (C2×Q8).138D4, D4.40(C4○D4), D4.D4⋊26C2, C4⋊C8.141C22, C4⋊C4.265C23, (C4×C8).301C22, (C2×C4).568C24, (C2×C8).372C23, D4.2D4.3C2, (C2×D8).94C22, C4⋊Q8.197C22, SD16⋊C4⋊48C2, C8⋊C4.67C22, C2.76(Q8⋊5D4), (C4×D4).206C22, (C2×D4).432C23, (C4×Q8).199C22, (C2×Q8).261C23, C4.Q8.183C22, C2.D8.137C22, C2.105(D4○SD16), D4⋊C4.91C22, (C2×SD16).73C22, C4.4D4.84C22, C22.828(C22×D4), C42.C2.69C22, Q8⋊C4.212C22, C2.103(D8⋊C22), C42.30C22⋊13C2, C22.50C24⋊12C2, C42.28C22⋊25C2, C42.78C22⋊24C2, C4.269(C2×C4○D4), (C2×C4).644(C2×D4), SmallGroup(128,2108)

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

Derived series
C1 — C2×C4 — C42.517C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — C42.517C23
Lower central
C1 — C2 — C2×C4 — C42.517C23
Upper central
C1 — C22 — C4×Q8 — C42.517C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.517C23

Subgroups: 320 in 173 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C8 [×4], C2×C4 [×7], C2×C4 [×12], D4 [×2], D4 [×3], Q8 [×6], C23 [×2], C42 [×3], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×7], C4⋊C4 [×9], C2×C8 [×4], D8 [×2], SD16 [×4], C22×C4 [×4], C2×D4 [×2], C2×Q8 [×3], C2×Q8, C4×C8, C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×6], C4⋊C8 [×3], C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4×Q8 [×3], C22⋊Q8 [×4], C4.4D4 [×2], C42.C2 [×2], C42⋊2C2 [×2], C4⋊Q8 [×2], C2×D8, C2×SD16 [×2], C4×SD16, SD16⋊C4, D8⋊C4, C8⋊4Q8, D4.D4, D4.2D4 [×2], Q8⋊Q8, D4.Q8, Q8.Q8, C42.78C22, C42.28C22, C42.30C22, D4⋊3Q8, C22.50C24, C42.517C23

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), Q8⋊5D4, D8⋊C22, D4○SD16, C42.517C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, 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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 63)(6 9 16 64)(7 10 13 61)(8 11 14 62)(29 36 37 41)(30 33 38 42)(31 34 39 43)(32 35 40 44)(45 51 56 60)(46 52 53 57)(47 49 54 58)(48 50 55 59)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

420000
0130000
0010150
0000161
0000160
0001160
,
100000
010000
0011500
0011600
0001601
00116160
,
190000
0160000
0013080
0013044
000040
0013440
,
400000
1130000
0013000
0001300
0013040
0013004
,
100000
010000
0011066
0014006
0001433
0014333

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,2,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,9,16,0,0,0,0,0,0,13,13,0,13,0,0,0,0,0,4,0,0,8,4,4,4,0,0,0,4,0,0],[4,1,0,0,0,0,0,13,0,0,0,0,0,0,13,0,13,13,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,14,0,14,0,0,0,0,14,3,0,0,6,0,3,3,0,0,6,6,3,3] >;
 

Character table of C42.517C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11114482222444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11111-1-1-1-1111-1-1111-1-1-1-111    linear of order 2
ρ31111-1-11-111-1-11-11-1-1-11-1-111-11-11-11    linear of order 2
ρ4111111-1-111-11-11-1-1-1-1-11-1111-11-1-11    linear of order 2
ρ51111-1-1111111111111-11-11-1-1-1-1-1-1-1    linear of order 2
ρ6111111-11111-1-1-1-11111-1-11-11111-1-1    linear of order 2
ρ71111111-111-1-11-11-1-1-1-1-111-11-11-11-1    linear of order 2
ρ81111-1-1-1-111-11-11-1-1-1-11111-1-11-111-1    linear of order 2
ρ9111111-1-111-1-11-1111-1-111-1-1-11-11-11    linear of order 2
ρ101111-1-11-111-11-11-111-11-11-1-11-11-1-11    linear of order 2
ρ111111-1-1-111111111-1-11-1-1-1-1-1111111    linear of order 2
ρ1211111111111-1-1-1-1-1-1111-1-1-1-1-1-1-111    linear of order 2
ρ131111-1-1-1-111-1-11-1111-111-1-111-11-11-1    linear of order 2
ρ141111111-111-11-11-111-1-1-1-1-11-11-111-1    linear of order 2
ρ15111111-111111111-1-111-11-11-1-1-1-1-1-1    linear of order 2
ρ161111-1-111111-1-1-1-1-1-11-111-111111-1-1    linear of order 2
ρ172222000-2-2-2-22-2-2200200000000000    orthogonal lifted from D4
ρ1822220002-2-22-2-22200-200000000000    orthogonal lifted from D4
ρ1922220002-2-2222-2-200-200000000000    orthogonal lifted from D4
ρ202222000-2-2-2-2-222-200200000000000    orthogonal lifted from D4
ρ212-22-2-2200-22000002i2i00000002i02i00    complex lifted from C4○D4
ρ222-22-22-200-22000002i2i00000002i02i00    complex lifted from C4○D4
ρ232-22-22-200-22000002i2i00000002i02i00    complex lifted from C4○D4
ρ242-22-2-2200-22000002i2i00000002i02i00    complex lifted from C4○D4
ρ254-44-400004-40000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-4-440004i004i000000000000000000    complex lifted from D8⋊C22
ρ274-4-440004i004i000000000000000000    complex lifted from D8⋊C22
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._{517}C_2^3
 
% in TeX
 
G:=Group("C4^2.517C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2108);
 
// by ID
 
G=gap.SmallGroup(128,2108);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,346,304,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,e*d*e=b^2*d>;
 
// generators/relations
 

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