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G = C42.28C23order 128 = 27

28th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.28C23, C4.362+ 1+4, C4.172- 1+4, C4⋊C4.143D4, Q8.Q833C2, C42Q1633C2, C4⋊C8.89C22, C22⋊C4.35D4, C2.35(Q8○D8), C8.D4.7C2, C23.96(C2×D4), C4⋊C4.200C23, (C2×C8).176C23, (C2×C4).459C24, C8.18D4.8C2, C4⋊Q8.130C22, C4.Q8.51C22, C2.D8.51C22, (C2×Q16).78C22, (C4×Q8).134C22, (C2×Q8).187C23, C22⋊Q8.53C22, (C22×C8).157C22, Q8⋊C4.62C22, C22.719(C22×D4), C42.C2.34C22, (C22×C4).1114C23, (C2×M4(2)).97C22, C42.6C22.4C2, C42⋊C2.177C22, C22.35C24.4C2, C2.78(C22.31C24), (C2×C4).583(C2×D4), SmallGroup(128,1993)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.28C23
C1C2C4C2×C4C42C4×Q8C22.35C24 — C42.28C23
C1C2C2×C4 — C42.28C23
C1C22C42⋊C2 — C42.28C23
C1C2C2C2×C4 — C42.28C23

Generators and relations for C42.28C23
 G = < a,b,c,d,e | a4=b4=1, c2=d2=e2=b2, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=a2d >

Subgroups: 268 in 161 conjugacy classes, 84 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×8], C23, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×16], C2×C8 [×4], C2×C8, M4(2), Q16 [×4], C22×C4, C2×Q8 [×4], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C42.C2 [×4], C422C2 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×Q16 [×4], C42.6C22, C42Q16 [×4], C8.18D4 [×2], C8.D4 [×2], Q8.Q8 [×4], C22.35C24 [×2], C42.28C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, Q8○D8 [×2], C42.28C23

Character table of C42.28C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114224444488888888444488
ρ111111111111111111111111111    trivial
ρ21111-111-1-111-11-111-1-11-11-11-11-1    linear of order 2
ρ311111111-1-1-1-1-1-1-11-1111-1-1-1-111    linear of order 2
ρ41111-111-11-1-11-11-111-11-1-11-111-1    linear of order 2
ρ51111-111-11-1-11-111-1-111-11-11-1-11    linear of order 2
ρ611111111-1-1-1-1-1-11-11-1111111-1-1    linear of order 2
ρ71111-111-1-111-11-1-1-1111-1-11-11-11    linear of order 2
ρ811111111111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ91111-111-1-111-1-1111-1-1-11-11-11-11    linear of order 2
ρ10111111111111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ111111-111-11-1-111-1-111-1-111-11-1-11    linear of order 2
ρ1211111111-1-1-1-111-11-11-1-11111-1-1    linear of order 2
ρ1311111111-1-1-1-1111-11-1-1-1-1-1-1-111    linear of order 2
ρ141111-111-11-1-111-11-1-11-11-11-111-1    linear of order 2
ρ15111111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ161111-111-1-111-1-11-1-111-111-11-11-1    linear of order 2
ρ172222-2-2-22-22-2200000000000000    orthogonal lifted from D4
ρ1822222-2-2-222-2-200000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-22200000000000000    orthogonal lifted from D4
ρ202222-2-2-222-22-200000000000000    orthogonal lifted from D4
ρ214-44-404-40000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-40-440000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2344-4-400000000000000000220-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-400000000000000000-2202200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-440000000000000000-22022000    symplectic lifted from Q8○D8, Schur index 2
ρ264-4-440000000000000000220-22000    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C42.28C23
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 23 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 64 9 15)(6 61 10 16)(7 62 11 13)(8 63 12 14)(29 35 37 41)(30 36 38 42)(31 33 39 43)(32 34 40 44)(45 57 51 53)(46 58 52 54)(47 59 49 55)(48 60 50 56)
(1 59 27 55)(2 58 28 54)(3 57 25 53)(4 60 26 56)(5 32 9 40)(6 31 10 39)(7 30 11 38)(8 29 12 37)(13 36 62 42)(14 35 63 41)(15 34 64 44)(16 33 61 43)(17 51 21 45)(18 50 22 48)(19 49 23 47)(20 52 24 46)
(1 19 27 23)(2 24 28 20)(3 17 25 21)(4 22 26 18)(5 64 9 15)(6 16 10 61)(7 62 11 13)(8 14 12 63)(29 33 37 43)(30 44 38 34)(31 35 39 41)(32 42 40 36)(45 59 51 55)(46 56 52 60)(47 57 49 53)(48 54 50 58)
(1 29 27 37)(2 30 28 38)(3 31 25 39)(4 32 26 40)(5 48 9 50)(6 45 10 51)(7 46 11 52)(8 47 12 49)(13 58 62 54)(14 59 63 55)(15 60 64 56)(16 57 61 53)(17 33 21 43)(18 34 22 44)(19 35 23 41)(20 36 24 42)

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

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

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

Matrix representation of C42.28C23 in GL8(𝔽17)

01000000
160000000
00010000
001600000
00000010
000016161615
00001000
00000001
,
160000000
016000000
001600000
000160000
00000100
000016000
000016161615
00001011
,
001100000
0010160000
167000000
71000000
0000121207
0000551010
00005500
0000120120
,
160000000
016000000
00100000
00010000
000001600
00001000
000016161615
00000111
,
00100000
00010000
10000000
01000000
00004448
000000130
000001300
000000013

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

C42.28C23 in GAP, Magma, Sage, TeX

C_4^2._{28}C_2^3
% in TeX

G:=Group("C4^2.28C2^3");
// GroupNames label

G:=SmallGroup(128,1993);
// by ID

G=gap.SmallGroup(128,1993);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,352,675,1018,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,e*d*e^-1=a^2*d>;
// generators/relations

Export

Character table of C42.28C23 in TeX

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