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G = C42.292C23order 128 = 27

153rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.292C23, C89D435C2, C86D436C2, C84Q835C2, C8.62(C4○D4), C4⋊D4.21C4, C22⋊Q8.21C4, C4⋊C8.234C22, (C4×M4(2))⋊36C2, (C2×C8).428C23, C42.215(C2×C4), (C4×C8).333C22, (C2×C4).664C24, C422C2.1C4, C4.4D4.17C4, (C4×D4).61C22, C42.C2.17C4, (C4×Q8).59C22, C82M4(2)⋊34C2, C42.6C447C2, C23.39(C22×C4), C22.D4.5C4, C8⋊C4.164C22, C2.22(Q8○M4(2)), C22⋊C8.141C22, (C22×C8).446C22, (C2×C42).774C22, C22.189(C23×C4), (C22×C4).935C23, C42.7C2223C2, C42⋊C2.307C22, (C2×M4(2)).366C22, C23.36C23.13C2, C2.46(C4×C4○D4), C4⋊C4.118(C2×C4), C4.315(C2×C4○D4), (C2×D4).141(C2×C4), C22⋊C4.17(C2×C4), (C2×Q8).120(C2×C4), (C22×C4).348(C2×C4), (C2×C4).272(C22×C4), SmallGroup(128,1699)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.292C23
C1C2C4C2×C4C42C4×C8C4×M4(2) — C42.292C23
C1C22 — C42.292C23
C1C2×C4 — C42.292C23
C1C2C2C2×C4 — C42.292C23

Generators and relations for C42.292C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=b2d >

Subgroups: 252 in 181 conjugacy classes, 128 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×8], D4 [×3], Q8, C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C4×C8 [×2], C8⋊C4 [×6], C22⋊C8 [×2], C22⋊C8 [×4], C4⋊C8 [×6], 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], C2×M4(2) [×4], C4×M4(2), C82M4(2) [×2], C42.6C4, C42.7C22 [×2], C89D4 [×4], C86D4 [×2], C84Q8 [×2], C23.36C23, C42.292C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], C4×C4○D4, Q8○M4(2) [×2], C42.292C23

Smallest permutation representation of C42.292C23
On 64 points
Generators in S64
(1 42 27 18)(2 23 28 47)(3 44 29 20)(4 17 30 41)(5 46 31 22)(6 19 32 43)(7 48 25 24)(8 21 26 45)(9 53 37 58)(10 63 38 50)(11 55 39 60)(12 57 40 52)(13 49 33 62)(14 59 34 54)(15 51 35 64)(16 61 36 56)
(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 6 3 8 5 2 7 4)(9 10 11 12 13 14 15 16)(17 46 19 48 21 42 23 44)(18 47 20 41 22 43 24 45)(25 30 27 32 29 26 31 28)(33 34 35 36 37 38 39 40)(49 63 51 57 53 59 55 61)(50 64 52 58 54 60 56 62)
(1 59)(2 55)(3 61)(4 49)(5 63)(6 51)(7 57)(8 53)(9 45)(10 22)(11 47)(12 24)(13 41)(14 18)(15 43)(16 20)(17 33)(19 35)(21 37)(23 39)(25 52)(26 58)(27 54)(28 60)(29 56)(30 62)(31 50)(32 64)(34 42)(36 44)(38 46)(40 48)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I···4Q8A···8H8I···8T
order1222222444444444···48···88···8
size1111444111122224···42···24···4

44 irreducible representations

dim11111111111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4○D4Q8○M4(2)
kernelC42.292C23C4×M4(2)C82M4(2)C42.6C4C42.7C22C89D4C86D4C84Q8C23.36C23C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C422C2C8C2
# reps11212422122422484

Matrix representation of C42.292C23 in GL6(𝔽17)

1590000
720000
004831
0001201
00125130
0001005
,
100000
010000
0013000
0001300
0000130
0000013
,
1600000
910000
00031610
000309
0044014
0008014
,
1600000
0160000
000319
000309
0013400
0008014
,
920000
1180000
008431
0012061
0051240
0010085

G:=sub<GL(6,GF(17))| [15,7,0,0,0,0,9,2,0,0,0,0,0,0,4,0,12,0,0,0,8,12,5,10,0,0,3,0,13,0,0,0,1,1,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,3,3,4,8,0,0,16,0,0,0,0,0,10,9,14,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,3,3,4,8,0,0,1,0,0,0,0,0,9,9,0,14],[9,11,0,0,0,0,2,8,0,0,0,0,0,0,8,12,5,10,0,0,4,0,12,0,0,0,3,6,4,8,0,0,1,1,0,5] >;

C42.292C23 in GAP, Magma, Sage, TeX

C_4^2._{292}C_2^3
% in TeX

G:=Group("C4^2.292C2^3");
// GroupNames label

G:=SmallGroup(128,1699);
// by ID

G=gap.SmallGroup(128,1699);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,2019,80,124]);
// Polycyclic

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

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