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G = C42.387C23order 128 = 27

248th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.387C23, (C4×D8)⋊40C2, C8⋊D48C2, C4⋊C4.247D4, (C4×Q16)⋊40C2, C82Q829C2, C8.10(C4○D4), C2.24(D4○D8), C87D4.11C2, C22⋊C4.87D4, C2.24(Q8○D8), C23.84(C2×D4), C8.18D436C2, C8.12D419C2, C4⋊C4.114C23, (C2×C4).373C24, (C2×C8).566C23, (C4×C8).223C22, (C4×D4).94C22, C4⋊Q8.116C22, SD16⋊C420C2, (C4×Q8).91C22, C82M4(2)⋊17C2, (C2×D8).134C22, (C2×D4).128C23, C4⋊D4.35C22, (C2×Q8).116C23, C8⋊C4.130C22, C2.D8.184C22, C22⋊Q8.35C22, (C22×C8).302C22, (C2×Q16).130C22, (C2×SD16).23C22, C4.4D4.35C22, C22.633(C22×D4), D4⋊C4.206C22, C22.36C245C2, (C22×C4).1053C23, Q8⋊C4.207C22, C42⋊C2.330C22, C42.28C2234C2, (C2×M4(2)).283C22, C2.70(C22.26C24), C4.58(C2×C4○D4), (C2×C4).145(C2×D4), SmallGroup(128,1907)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.387C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.387C23
C1C2C2×C4 — C42.387C23
C1C22C42⋊C2 — C42.387C23
C1C2C2C2×C4 — C42.387C23

Generators and relations for C42.387C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=cbc=b-1, ab=ba, ac=ca, dad=ab2, ae=ea, bd=db, be=eb, dcd=a2c, ece-1=b-1c, de=ed >

Subgroups: 348 in 181 conjugacy classes, 88 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×6], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4.4D4 [×2], C422C2 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×Q16, C82M4(2), C4×D8, C4×Q16, SD16⋊C4 [×2], C87D4, C8.18D4, C8⋊D4 [×2], C42.28C22 [×2], C8.12D4, C82Q8, C22.36C24 [×2], C42.387C23
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, Q8○D8, C42.387C23

Smallest permutation representation of C42.387C23
On 64 points
Generators in S64
(1 35 27 10)(2 36 28 11)(3 37 29 12)(4 38 30 13)(5 39 31 14)(6 40 32 15)(7 33 25 16)(8 34 26 9)(17 58 45 54)(18 59 46 55)(19 60 47 56)(20 61 48 49)(21 62 41 50)(22 63 42 51)(23 64 43 52)(24 57 44 53)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 48)(24 47)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 40)(38 39)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(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)| (1,35,27,10)(2,36,28,11)(3,37,29,12)(4,38,30,13)(5,39,31,14)(6,40,32,15)(7,33,25,16)(8,34,26,9)(17,58,45,54)(18,59,46,55)(19,60,47,56)(20,61,48,49)(21,62,41,50)(22,63,42,51)(23,64,43,52)(24,57,44,53), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,48)(24,47)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(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( (1,35,27,10)(2,36,28,11)(3,37,29,12)(4,38,30,13)(5,39,31,14)(6,40,32,15)(7,33,25,16)(8,34,26,9)(17,58,45,54)(18,59,46,55)(19,60,47,56)(20,61,48,49)(21,62,41,50)(22,63,42,51)(23,64,43,52)(24,57,44,53), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,48)(24,47)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(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([(1,35,27,10),(2,36,28,11),(3,37,29,12),(4,38,30,13),(5,39,31,14),(6,40,32,15),(7,33,25,16),(8,34,26,9),(17,58,45,54),(18,59,46,55),(19,60,47,56),(20,61,48,49),(21,62,41,50),(22,63,42,51),(23,64,43,52),(24,57,44,53)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,48),(24,47),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35),(37,40),(38,39),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(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 2A2B2C2D2E2F4A···4F4G4H4I4J···4O8A8B8C8D8E···8J
order12222224···44444···488888···8
size11114882···24448···822224···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8Q8○D8
kernelC42.387C23C82M4(2)C4×D8C4×Q16SD16⋊C4C87D4C8.18D4C8⋊D4C42.28C22C8.12D4C82Q8C22.36C24C22⋊C4C4⋊C4C8C2C2
# reps11112112211222822

Matrix representation of C42.387C23 in GL6(𝔽17)

400000
040000
00160150
00016015
001010
000101
,
100000
010000
000100
0016000
000001
0000160
,
1600000
810000
0031400
00141400
0000314
00001414
,
1130000
0160000
0001309
004080
000004
0000130
,
1600000
0160000
0031400
003300
0000314
000033

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

C42.387C23 in GAP, Magma, Sage, TeX

C_4^2._{387}C_2^3
% in TeX

G:=Group("C4^2.387C2^3");
// GroupNames label

G:=SmallGroup(128,1907);
// by ID

G=gap.SmallGroup(128,1907);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,520,1018,80,4037,124]);
// Polycyclic

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

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