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G = C42.391C23order 128 = 27

252nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.391C23, C8⋊Q89C2, (C4×D8)⋊30C2, C8⋊D49C2, C82D47C2, C83D47C2, C87D427C2, C88D415C2, C4⋊C4.251D4, D8⋊C415C2, (C4×SD16)⋊18C2, C8.30(C4○D4), C2.26(D4○D8), C4.4D842C2, C22⋊C4.91D4, C23.88(C2×D4), C4⋊C4.118C23, (C2×C4).377C24, (C4×C8).184C22, (C2×C8).279C23, (C4×D4).97C22, C4⋊Q8.119C22, SD16⋊C422C2, (C4×Q8).94C22, C82M4(2)⋊20C2, C4.Q8.29C22, C2.40(D4○SD16), (C2×D8).164C22, (C2×D4).131C23, C4⋊D4.38C22, C41D4.66C22, (C2×Q8).119C23, C2.D8.221C22, C8⋊C4.134C22, C22⋊Q8.38C22, (C22×C8).279C22, (C2×SD16).25C22, C4.4D4.37C22, C22.637(C22×D4), C42.C2.23C22, D4⋊C4.149C22, (C22×C4).1057C23, C22.36C246C2, C22.34C244C2, Q8⋊C4.141C22, C42.78C2230C2, C42⋊C2.334C22, (C2×M4(2)).287C22, C2.74(C22.26C24), C4.62(C2×C4○D4), (C2×C4).149(C2×D4), SmallGroup(128,1911)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.391C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.391C23
C1C2C2×C4 — C42.391C23
C1C22C42⋊C2 — C42.391C23
C1C2C2C2×C4 — C42.391C23

Generators and relations for C42.391C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=b2, e2=b, ab=ba, ac=ca, dad=ab2, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd=a2c, ece-1=b-1c, ede-1=b2d >

Subgroups: 388 in 187 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×11], Q8 [×3], C23, C23 [×3], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×5], C2×Q8, C2×Q8, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×3], C4.4D4, C4.4D4, C42.C2, C422C2, C41D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C82M4(2), C4×D8, C4×SD16, SD16⋊C4, D8⋊C4, C88D4, C87D4, C8⋊D4, C82D4, C4.4D8, C42.78C22, C83D4, C8⋊Q8, C22.34C24, C22.36C24, C42.391C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D4○D8, D4○SD16, C42.391C23

Smallest permutation representation of C42.391C23
On 64 points
Generators in S64
(1 38 31 10)(2 39 32 11)(3 40 25 12)(4 33 26 13)(5 34 27 14)(6 35 28 15)(7 36 29 16)(8 37 30 9)(17 57 50 42)(18 58 51 43)(19 59 52 44)(20 60 53 45)(21 61 54 46)(22 62 55 47)(23 63 56 48)(24 64 49 41)
(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 56 5 52)(2 51 6 55)(3 54 7 50)(4 49 8 53)(9 60 13 64)(10 63 14 59)(11 58 15 62)(12 61 16 57)(17 25 21 29)(18 28 22 32)(19 31 23 27)(20 26 24 30)(33 41 37 45)(34 44 38 48)(35 47 39 43)(36 42 40 46)
(1 44)(2 41)(3 46)(4 43)(5 48)(6 45)(7 42)(8 47)(9 51)(10 56)(11 53)(12 50)(13 55)(14 52)(15 49)(16 54)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 61)(26 58)(27 63)(28 60)(29 57)(30 62)(31 59)(32 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)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J···4N8A8B8C8D8E···8J
order122222224···44444···488888···8
size111148882···24448···822224···4

32 irreducible representations

dim111111111111111122244
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8D4○SD16
kernelC42.391C23C82M4(2)C4×D8C4×SD16SD16⋊C4D8⋊C4C88D4C87D4C8⋊D4C82D4C4.4D8C42.78C22C83D4C8⋊Q8C22.34C24C22.36C24C22⋊C4C4⋊C4C8C2C2
# reps111111111111111122822

Matrix representation of C42.391C23 in GL6(𝔽17)

1300000
0130000
000010
000001
001000
000100
,
100000
010000
000100
0016000
000001
0000160
,
1390000
440000
00120012
0005120
00012120
0012005
,
120000
0160000
00012120
0050012
005005
0005120
,
100000
010000
0000125
00001212
0012500
00121200

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,12,0,0,12,0,0,0,5,12,0,0,0,0,12,12,0,0,0,12,0,0,5],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,5,5,0,0,0,12,0,0,5,0,0,12,0,0,12,0,0,0,12,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,12,12,0,0,0,0,5,12,0,0] >;

C42.391C23 in GAP, Magma, Sage, TeX

C_4^2._{391}C_2^3
% in TeX

G:=Group("C4^2.391C2^3");
// GroupNames label

G:=SmallGroup(128,1911);
// by ID

G=gap.SmallGroup(128,1911);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,184,1018,80,4037,124]);
// Polycyclic

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

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