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G = C42.291C23order 128 = 27

152nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.291C23, (C8×D4)⋊41C2, (C8×Q8)⋊30C2, C83(C4⋊D4), C82(C89D4), C82(C86D4), C89D454C2, C86D452C2, C82(C84Q8), C84Q852C2, C83(C22⋊Q8), C8(C422C2), C4.20(C8○D4), C82(C4.4D4), C4⋊D4.36C4, C82(C42.C2), C22⋊Q8.36C4, C8.106(C4○D4), C4⋊C8.362C22, C8(C42.6C4), (C2×C4).663C24, C422C2.8C4, C42.287(C2×C4), (C4×C8).438C22, (C2×C8).641C23, C4.4D4.28C4, C22.3(C8○D4), C42.C2.28C4, C82M4(2)⋊33C2, C42.6C459C2, (C4×D4).294C22, C82(C22.D4), C23.38(C22×C4), (C4×Q8).279C22, C8⋊C4.163C22, C22⋊C8.233C22, C8(C42.7C22), (C22×C8).516C22, C22.188(C23×C4), C22.D4.16C4, C8(C23.36C23), (C2×C42).1120C22, C42.7C2234C2, (C22×C4).1278C23, C42⋊C2.306C22, (C2×M4(2)).365C22, C23.36C23.35C2, (C2×C4×C8)⋊45C2, (C2×C8)(C84Q8), C2.24(C2×C8○D4), C2.45(C4×C4○D4), C4⋊C4.164(C2×C4), (C2×C8)(C4.4D4), C4.314(C2×C4○D4), (C2×C8)(C42.C2), (C2×C8)(C422C2), (C2×D4).180(C2×C4), C22⋊C4.41(C2×C4), (C2×C4).77(C22×C4), (C2×Q8).164(C2×C4), (C2×C8)(C42.6C4), (C22×C4).388(C2×C4), (C2×C8)(C22.D4), (C2×C8)(C42.7C22), (C2×C8)(C23.36C23), SmallGroup(128,1698)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.291C23
C1C2C4C2×C4C42C4×C8C2×C4×C8 — C42.291C23
C1C22 — C42.291C23
C1C2×C8 — C42.291C23
C1C2C2C2×C4 — C42.291C23

Generators and relations for C42.291C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede-1=b2d >

Subgroups: 252 in 190 conjugacy classes, 132 normal (52 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×4], C4×C8 [×2], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×4], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4×C8, C82M4(2) [×2], C42.6C4, C42.7C22 [×2], C8×D4, C8×D4 [×2], C89D4 [×2], C86D4, C8×Q8, C84Q8, C23.36C23, C42.291C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C8○D4 [×4], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C2×C8○D4 [×2], C42.291C23

Smallest permutation representation of C42.291C23
On 64 points
Generators in S64
(1 33 57 48)(2 34 58 41)(3 35 59 42)(4 36 60 43)(5 37 61 44)(6 38 62 45)(7 39 63 46)(8 40 64 47)(9 51 28 17)(10 52 29 18)(11 53 30 19)(12 54 31 20)(13 55 32 21)(14 56 25 22)(15 49 26 23)(16 50 27 24)
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(17 49 21 53)(18 50 22 54)(19 51 23 55)(20 52 24 56)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 21 13 17)(10 52 14 56)(11 23 15 19)(12 54 16 50)(18 25 22 29)(20 27 24 31)(26 53 30 49)(28 55 32 51)(33 61 37 57)(35 63 39 59)(41 58 45 62)(43 60 47 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 61 54)(2 21 62 51)(3 18 63 56)(4 23 64 53)(5 20 57 50)(6 17 58 55)(7 22 59 52)(8 19 60 49)(9 45 32 34)(10 42 25 39)(11 47 26 36)(12 44 27 33)(13 41 28 38)(14 46 29 35)(15 43 30 40)(16 48 31 37)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8H8I···8T8U···8AB
order1222222244444···44···48···88···88···8
size1111224411112···24···41···12···24···4

56 irreducible representations

dim11111111111111111222
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4○D4C8○D4C8○D4
kernelC42.291C23C2×C4×C8C82M4(2)C42.6C4C42.7C22C8×D4C89D4C86D4C8×Q8C84Q8C23.36C23C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C422C2C8C4C22
# reps11212321111224224888

Matrix representation of C42.291C23 in GL4(𝔽17) generated by

11600
21600
00015
0080
,
1000
0100
00130
00013
,
16000
15100
0002
0080
,
16100
15100
00013
00160
,
4000
0400
00015
0090
G:=sub<GL(4,GF(17))| [1,2,0,0,16,16,0,0,0,0,0,8,0,0,15,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[16,15,0,0,0,1,0,0,0,0,0,8,0,0,2,0],[16,15,0,0,1,1,0,0,0,0,0,16,0,0,13,0],[4,0,0,0,0,4,0,0,0,0,0,9,0,0,15,0] >;

C42.291C23 in GAP, Magma, Sage, TeX

C_4^2._{291}C_2^3
% in TeX

G:=Group("C4^2.291C2^3");
// GroupNames label

G:=SmallGroup(128,1698);
// by ID

G=gap.SmallGroup(128,1698);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,80,124]);
// Polycyclic

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

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