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G = C42.695C23order 128 = 27

110th non-split extension by C42 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.695C23, C4.1172- 1+4, (C8×Q8)⋊3C2, (C2×Q8)⋊7C8, Q8.8(C2×C8), Q82(C22⋊C8), C2.8(C23×C8), (C4×Q8).30C4, C4.20(C22×C8), (C4×C8).30C22, C4⋊C8.376C22, (C2×C4).679C24, (C2×C8).482C23, C42.227(C2×C4), (C22×Q8).32C4, C2.4(Q8○M4(2)), C22.15(C22×C8), C22.44(C23×C4), (C4×Q8).333C22, C22⋊C8.245C22, (C2×C42).786C22, C23.230(C22×C4), C42.12C4.32C2, (C22×C4).1283C23, C2.3(C23.32C23), C22⋊C8(C4×Q8), (C2×C4⋊C4).79C4, (C2×C4×Q8).45C2, (C2×C4).32(C2×C8), C4⋊C4.251(C2×C4), (C2×Q8).229(C2×C4), (C22×C4).140(C2×C4), (C2×C4).475(C22×C4), SmallGroup(128,1714)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C42.695C23
C1C2C4C2×C4C22×C4C2×C42C2×C4×Q8 — C42.695C23
C1C2 — C42.695C23
C1C2×C4 — C42.695C23
C1C2C2C2×C4 — C42.695C23

Generators and relations for C42.695C23
 G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >

Subgroups: 228 in 198 conjugacy classes, 174 normal (12 characteristic)
C1, C2 [×3], C2 [×2], C4 [×14], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×24], C2×C4 [×5], Q8 [×16], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×8], C22×C4, C22×C4 [×6], C2×Q8 [×12], C4×C8 [×12], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C42.12C4 [×6], C8×Q8 [×8], C2×C4×Q8, C42.695C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2- 1+4 [×2], C23.32C23, C23×C8, Q8○M4(2), C42.695C23

Smallest permutation representation of C42.695C23
On 64 points
Generators in S64
(1 35 55 45)(2 36 56 46)(3 37 49 47)(4 38 50 48)(5 39 51 41)(6 40 52 42)(7 33 53 43)(8 34 54 44)(9 32 64 18)(10 25 57 19)(11 26 58 20)(12 27 59 21)(13 28 60 22)(14 29 61 23)(15 30 62 24)(16 31 63 17)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 31 55 17)(2 32 56 18)(3 25 49 19)(4 26 50 20)(5 27 51 21)(6 28 52 22)(7 29 53 23)(8 30 54 24)(9 46 64 36)(10 47 57 37)(11 48 58 38)(12 41 59 39)(13 42 60 40)(14 43 61 33)(15 44 62 34)(16 45 63 35)
(1 45 55 35)(2 36 56 46)(3 47 49 37)(4 38 50 48)(5 41 51 39)(6 40 52 42)(7 43 53 33)(8 34 54 44)(9 32 64 18)(10 19 57 25)(11 26 58 20)(12 21 59 27)(13 28 60 22)(14 23 61 29)(15 30 62 24)(16 17 63 31)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4AD8A···8AF
order12222244444···48···8
size11112211112···22···2

68 irreducible representations

dim1111111144
type++++-
imageC1C2C2C2C4C4C4C82- 1+4Q8○M4(2)
kernelC42.695C23C42.12C4C8×Q8C2×C4×Q8C2×C4⋊C4C4×Q8C22×Q8C2×Q8C4C2
# reps16816823222

Matrix representation of C42.695C23 in GL5(𝔽17)

160000
05500
051200
00055
000512
,
130000
04000
00400
00040
00004
,
90000
020150
002015
000150
000015
,
160000
00100
016000
00001
000160
,
10000
0121200
012500
07755
0710512

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,5,5,0,0,0,5,12,0,0,0,0,0,5,5,0,0,0,5,12],[13,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[9,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,15,0,15,0,0,0,15,0,15],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,12,12,7,7,0,12,5,7,10,0,0,0,5,5,0,0,0,5,12] >;

C42.695C23 in GAP, Magma, Sage, TeX

C_4^2._{695}C_2^3
% in TeX

G:=Group("C4^2.695C2^3");
// GroupNames label

G:=SmallGroup(128,1714);
// by ID

G=gap.SmallGroup(128,1714);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,219,100,675,124]);
// Polycyclic

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

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