Copied to
clipboard

G = C23.41D8order 128 = 27

12nd non-split extension by C23 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.41D8, C8.95(C2×D4), (C2×Q16)⋊14C4, (C2×C4).139D8, (C2×C8).119D4, Q16.8(C2×C4), C8.6(C22⋊C4), C8.32(C22×C4), C4.13(C2×SD16), (C2×C4).49SD16, C22.56(C2×D8), C2.Q3215C2, (C2×C8).497C23, (C2×C16).51C22, C2.2(Q32⋊C2), (C22×C4).332D4, C4.12(D4⋊C4), (C2×M5(2)).21C2, (C22×Q16).13C2, C2.D8.145C22, (C22×C8).231C22, (C2×Q16).101C22, C22.30(D4⋊C4), C23.25D4.14C2, (C2×C8).81(C2×C4), (C2×C4).759(C2×D4), C4.53(C2×C22⋊C4), C2.31(C2×D4⋊C4), (C2×C4).150(C22⋊C4), SmallGroup(128,873)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.41D8
C1C2C4C2×C4C2×C8C22×C8C22×Q16 — C23.41D8
C1C2C4C8 — C23.41D8
C1C22C22×C4C22×C8 — C23.41D8
C1C2C2C2C2C4C4C2×C8 — C23.41D8

Generators and relations for C23.41D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd7 >

Subgroups: 228 in 110 conjugacy classes, 52 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], Q16 [×4], Q16 [×6], C22×C4, C22×C4, C2×Q8 [×9], C4.Q8, C2.D8 [×2], C2×C16 [×2], M5(2) [×2], C42⋊C2, C22×C8, C2×Q16 [×6], C2×Q16 [×3], C22×Q8, C2.Q32 [×4], C23.25D4, C2×M5(2), C22×Q16, C23.41D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, Q32⋊C2 [×2], C23.41D8

Smallest permutation representation of C23.41D8
On 64 points
Generators in S64
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 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 18 60 40)(2 47 61 25)(3 32 62 38)(4 45 63 23)(5 30 64 36)(6 43 49 21)(7 28 50 34)(8 41 51 19)(9 26 52 48)(10 39 53 17)(11 24 54 46)(12 37 55 31)(13 22 56 44)(14 35 57 29)(15 20 58 42)(16 33 59 27)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4L8A8B8C8D8E8F16A···16H
order12222244444···488888816···16
size11112222228···82222444···4

32 irreducible representations

dim111111222224
type+++++++++-
imageC1C2C2C2C2C4D4D4D8SD16D8Q32⋊C2
kernelC23.41D8C2.Q32C23.25D4C2×M5(2)C22×Q16C2×Q16C2×C8C22×C4C2×C4C2×C4C23C2
# reps141118312424

Matrix representation of C23.41D8 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
2100000
1350000
0000710
00001210
0014300
008300
,
520000
4120000
000010
000001
0016000
0001600

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

C23.41D8 in GAP, Magma, Sage, TeX

C_2^3._{41}D_8
% in TeX

G:=Group("C2^3.41D8");
// GroupNames label

G:=SmallGroup(128,873);
// by ID

G=gap.SmallGroup(128,873);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^7>;
// generators/relations

׿
×
𝔽