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G = C42.28C23  order 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.Q8⋊33C2, C4⋊2Q16⋊33C2, 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, C2.D8.51C22, C4.Q8.51C22, (C2×Q8).187C23, (C2×Q16).78C22, (C4×Q8).134C22, 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

Derived series
C1 — C2×C4 — C42.28C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C22.35C24 — C42.28C23
Lower central
C1 — C2 — C2×C4 — C42.28C23
Upper central
C1 — C22 — C42⋊C2 — C42.28C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.28C23

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], C42⋊2C2 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×Q16 [×4], C42.6C22, C4⋊2Q16 [×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

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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
ρ2344-4-4000000000000000002√202√200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-4000000000000000002√202√200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ264-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2

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
 

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