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G = C42.389C23order 128 = 27

250th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.389C23, C4⋊C4.249D4, (C4×Q16)⋊41C2, C4⋊Q1620C2, C8.12(C4○D4), C22⋊C4.89D4, C2.25(Q8○D8), C8.5Q819C2, C8.D4.4C2, C23.86(C2×D4), Q16⋊C414C2, C4⋊C4.116C23, (C4×C8).225C22, (C2×C4).375C24, (C2×C8).568C23, C4⋊Q8.117C22, (C4×Q8).92C22, C4.Q8.27C22, C8.18D4.11C2, (C2×Q16).65C22, (C2×Q8).117C23, C2.D8.186C22, C8⋊C4.132C22, C22⋊Q8.36C22, C82M4(2).11C2, (C22×C8).304C22, C22.635(C22×D4), C42.C2.21C22, (C22×C4).1055C23, Q8⋊C4.208C22, C42⋊C2.332C22, C42.30C2221C2, (C2×M4(2)).285C22, C22.35C24.2C2, C2.72(C22.26C24), C4.60(C2×C4○D4), (C2×C4).147(C2×D4), SmallGroup(128,1909)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.389C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.389C23
C1C2C2×C4 — C42.389C23
C1C22C42⋊C2 — C42.389C23
C1C2C2C2×C4 — C42.389C23

Generators and relations for C42.389C23
 G = < a,b,c,d,e | a4=b4=1, c2=d2=b2, e2=cbc-1=b-1, ab=ba, ac=ca, dad-1=ab2, ae=ea, bd=db, be=eb, dcd-1=a2c, ece-1=b-1c, de=ed >

Subgroups: 268 in 169 conjugacy classes, 88 normal (26 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×8], C23, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×16], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], Q16 [×8], C22×C4, C2×Q8 [×4], C4×C8 [×2], C8⋊C4 [×2], Q8⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C42.C2 [×4], C422C2 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×Q16 [×4], C82M4(2), C4×Q16 [×2], Q16⋊C4 [×2], C8.18D4 [×2], C8.D4 [×2], C42.30C22 [×2], C4⋊Q16, C8.5Q8, C22.35C24 [×2], C42.389C23
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, Q8○D8 [×2], C42.389C23

Smallest permutation representation of C42.389C23
On 64 points
Generators in S64
(1 33 27 16)(2 34 28 9)(3 35 29 10)(4 36 30 11)(5 37 31 12)(6 38 32 13)(7 39 25 14)(8 40 26 15)(17 62 53 42)(18 63 54 43)(19 64 55 44)(20 57 56 45)(21 58 49 46)(22 59 50 47)(23 60 51 48)(24 61 52 41)
(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 45 5 41)(2 44 6 48)(3 43 7 47)(4 42 8 46)(9 55 13 51)(10 54 14 50)(11 53 15 49)(12 52 16 56)(17 40 21 36)(18 39 22 35)(19 38 23 34)(20 37 24 33)(25 59 29 63)(26 58 30 62)(27 57 31 61)(28 64 32 60)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 59 45 63)(42 60 46 64)(43 61 47 57)(44 62 48 58)
(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)| (1,33,27,16)(2,34,28,9)(3,35,29,10)(4,36,30,11)(5,37,31,12)(6,38,32,13)(7,39,25,14)(8,40,26,15)(17,62,53,42)(18,63,54,43)(19,64,55,44)(20,57,56,45)(21,58,49,46)(22,59,50,47)(23,60,51,48)(24,61,52,41), (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), (1,45,5,41)(2,44,6,48)(3,43,7,47)(4,42,8,46)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,55,21,51)(18,56,22,52)(19,49,23,53)(20,50,24,54)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58), (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( (1,33,27,16)(2,34,28,9)(3,35,29,10)(4,36,30,11)(5,37,31,12)(6,38,32,13)(7,39,25,14)(8,40,26,15)(17,62,53,42)(18,63,54,43)(19,64,55,44)(20,57,56,45)(21,58,49,46)(22,59,50,47)(23,60,51,48)(24,61,52,41), (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), (1,45,5,41)(2,44,6,48)(3,43,7,47)(4,42,8,46)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,55,21,51)(18,56,22,52)(19,49,23,53)(20,50,24,54)(25,27,29,31)(26,28,30,32)(33,39,37,35)(34,40,38,36)(41,59,45,63)(42,60,46,64)(43,61,47,57)(44,62,48,58), (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([(1,33,27,16),(2,34,28,9),(3,35,29,10),(4,36,30,11),(5,37,31,12),(6,38,32,13),(7,39,25,14),(8,40,26,15),(17,62,53,42),(18,63,54,43),(19,64,55,44),(20,57,56,45),(21,58,49,46),(22,59,50,47),(23,60,51,48),(24,61,52,41)], [(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)], [(1,45,5,41),(2,44,6,48),(3,43,7,47),(4,42,8,46),(9,55,13,51),(10,54,14,50),(11,53,15,49),(12,52,16,56),(17,40,21,36),(18,39,22,35),(19,38,23,34),(20,37,24,33),(25,59,29,63),(26,58,30,62),(27,57,31,61),(28,64,32,60)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,55,21,51),(18,56,22,52),(19,49,23,53),(20,50,24,54),(25,27,29,31),(26,28,30,32),(33,39,37,35),(34,40,38,36),(41,59,45,63),(42,60,46,64),(43,61,47,57),(44,62,48,58)], [(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 2A2B2C2D4A···4F4G4H4I4J···4Q8A8B8C8D8E···8J
order122224···44444···488888···8
size111142···24448···822224···4

32 irreducible representations

dim11111111112224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4Q8○D8
kernelC42.389C23C82M4(2)C4×Q16Q16⋊C4C8.18D4C8.D4C42.30C22C4⋊Q16C8.5Q8C22.35C24C22⋊C4C4⋊C4C8C2
# reps11222221122284

Matrix representation of C42.389C23 in GL6(𝔽17)

400000
040000
000010
000001
001000
000100
,
100000
010000
000100
0016000
000001
0000160
,
13150000
1640000
00001212
0000125
00121200
0012500
,
100000
13160000
0001600
001000
000001
0000160
,
100000
010000
0014300
00141400
0000143
00001414

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

C42.389C23 in GAP, Magma, Sage, TeX

C_4^2._{389}C_2^3
% in TeX

G:=Group("C4^2.389C2^3");
// GroupNames label

G:=SmallGroup(128,1909);
// by ID

G=gap.SmallGroup(128,1909);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,520,1018,80,4037,124]);
// Polycyclic

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

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