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G = C42.386C23order 128 = 27

247th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.386C23, C8⋊D47C2, C85D46C2, C88D446C2, C4⋊C4.246D4, C8.9(C4○D4), (C4×SD16)⋊54C2, C22⋊C4.86D4, C8.5Q817C2, C23.83(C2×D4), C4⋊C4.113C23, (C2×C8).601C23, (C4×C8).291C22, (C2×C4).372C24, (C4×D4).93C22, C4⋊Q8.115C22, SD16⋊C419C2, (C4×Q8).90C22, C82M4(2)⋊16C2, C2.D8.96C22, C2.38(D4○SD16), (C2×D4).127C23, C4⋊D4.34C22, C41D4.64C22, (C2×Q8).115C23, C8⋊C4.129C22, C4.Q8.164C22, C22⋊Q8.34C22, (C22×C8).360C22, (C2×SD16).22C22, C22.632(C22×D4), C42.C2.19C22, D4⋊C4.205C22, C22.35C243C2, (C22×C4).1052C23, Q8⋊C4.128C22, C42⋊C2.329C22, C42.30C2220C2, C42.29C2220C2, (C2×M4(2)).282C22, C22.34C24.2C2, C2.69(C22.26C24), C4.57(C2×C4○D4), (C2×C4).144(C2×D4), SmallGroup(128,1906)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.386C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.386C23
C1C2C2×C4 — C42.386C23
C1C22C42⋊C2 — C42.386C23
C1C2C2C2×C4 — C42.386C23

Generators and relations for C42.386C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=b2, e2=b, ab=ba, ac=ca, dad-1=ab2, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a2c, ece-1=b-1c, de=ed >

Subgroups: 348 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×2], C42.C2 [×2], C422C2 [×2], C41D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×4], C82M4(2), C4×SD16 [×2], SD16⋊C4 [×2], C88D4 [×2], C8⋊D4 [×2], C42.29C22, C42.30C22, C85D4, C8.5Q8, C22.34C24, C22.35C24, C42.386C23
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○SD16 [×2], C42.386C23

Smallest permutation representation of C42.386C23
On 64 points
Generators in S64
(1 39 31 14)(2 40 32 15)(3 33 25 16)(4 34 26 9)(5 35 27 10)(6 36 28 11)(7 37 29 12)(8 38 30 13)(17 57 44 53)(18 58 45 54)(19 59 46 55)(20 60 47 56)(21 61 48 49)(22 62 41 50)(23 63 42 51)(24 64 43 52)
(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 46)(2 41)(3 44)(4 47)(5 42)(6 45)(7 48)(8 43)(9 60)(10 63)(11 58)(12 61)(13 64)(14 59)(15 62)(16 57)(17 25)(18 28)(19 31)(20 26)(21 29)(22 32)(23 27)(24 30)(33 53)(34 56)(35 51)(36 54)(37 49)(38 52)(39 55)(40 50)
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)
(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)| (1,39,31,14)(2,40,32,15)(3,33,25,16)(4,34,26,9)(5,35,27,10)(6,36,28,11)(7,37,29,12)(8,38,30,13)(17,57,44,53)(18,58,45,54)(19,59,46,55)(20,60,47,56)(21,61,48,49)(22,62,41,50)(23,63,42,51)(24,64,43,52), (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), (1,46)(2,41)(3,44)(4,47)(5,42)(6,45)(7,48)(8,43)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,25)(18,28)(19,31)(20,26)(21,29)(22,32)(23,27)(24,30)(33,53)(34,56)(35,51)(36,54)(37,49)(38,52)(39,55)(40,50), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (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( (1,39,31,14)(2,40,32,15)(3,33,25,16)(4,34,26,9)(5,35,27,10)(6,36,28,11)(7,37,29,12)(8,38,30,13)(17,57,44,53)(18,58,45,54)(19,59,46,55)(20,60,47,56)(21,61,48,49)(22,62,41,50)(23,63,42,51)(24,64,43,52), (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), (1,46)(2,41)(3,44)(4,47)(5,42)(6,45)(7,48)(8,43)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,25)(18,28)(19,31)(20,26)(21,29)(22,32)(23,27)(24,30)(33,53)(34,56)(35,51)(36,54)(37,49)(38,52)(39,55)(40,50), (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,45,13,41)(10,46,14,42)(11,47,15,43)(12,48,16,44)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,61,29,57)(26,62,30,58)(27,63,31,59)(28,64,32,60), (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([(1,39,31,14),(2,40,32,15),(3,33,25,16),(4,34,26,9),(5,35,27,10),(6,36,28,11),(7,37,29,12),(8,38,30,13),(17,57,44,53),(18,58,45,54),(19,59,46,55),(20,60,47,56),(21,61,48,49),(22,62,41,50),(23,63,42,51),(24,64,43,52)], [(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)], [(1,46),(2,41),(3,44),(4,47),(5,42),(6,45),(7,48),(8,43),(9,60),(10,63),(11,58),(12,61),(13,64),(14,59),(15,62),(16,57),(17,25),(18,28),(19,31),(20,26),(21,29),(22,32),(23,27),(24,30),(33,53),(34,56),(35,51),(36,54),(37,49),(38,52),(39,55),(40,50)], [(1,55,5,51),(2,56,6,52),(3,49,7,53),(4,50,8,54),(9,45,13,41),(10,46,14,42),(11,47,15,43),(12,48,16,44),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,61,29,57),(26,62,30,58),(27,63,31,59),(28,64,32,60)], [(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

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16
kernelC42.386C23C82M4(2)C4×SD16SD16⋊C4C88D4C8⋊D4C42.29C22C42.30C22C85D4C8.5Q8C22.34C24C22.35C24C22⋊C4C4⋊C4C8C2
# reps1122221111112284

Matrix representation of C42.386C23 in GL6(𝔽17)

400000
040000
000010
000001
0016000
0001600
,
1600000
0160000
000100
0016000
000001
0000160
,
1150000
0160000
0001450
00140012
00120014
0005140
,
490000
4130000
00120014
0001230
0001450
003005
,
1150000
1160000
0051200
005500
0000512
000055

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

C42.386C23 in GAP, Magma, Sage, TeX

C_4^2._{386}C_2^3
% in TeX

G:=Group("C4^2.386C2^3");
// GroupNames label

G:=SmallGroup(128,1906);
// by ID

G=gap.SmallGroup(128,1906);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,184,1018,80,4037,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=b^2,e^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=b^-1,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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