Copied to
clipboard

G = C24.67D4order 128 = 27

22nd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.67D4, C81(C22⋊C4), (C2×C8).100D4, C2.1(C8⋊D4), C2.1(C82D4), (C2×M4(2))⋊5C4, (C22×C4).43Q8, C23.26(C4⋊C4), C2.1(C8.D4), C23.746(C2×D4), (C22×C4).271D4, C4.71(C22⋊Q8), C22.4Q1646C2, C4.54(C42⋊C2), C23.7Q8.8C2, C22.62(C8⋊C22), (C22×C8).212C22, (C23×C4).237C22, (C22×M4(2)).1C2, C22.111(C4⋊D4), C2.7(M4(2)⋊C4), (C22×C4).1331C23, C22.51(C8.C22), C2.18(C23.7Q8), (C2×C4.Q8)⋊1C2, (C2×C2.D8)⋊32C2, (C2×C4).42(C4⋊C4), (C2×C8).142(C2×C4), C4.88(C2×C22⋊C4), C22.92(C2×C4⋊C4), (C2×C4).189(C2×Q8), (C2×C4).1320(C2×D4), (C2×C4⋊C4).38C22, (C2×C4).553(C4○D4), (C22×C4).264(C2×C4), (C2×C4).529(C22×C4), SmallGroup(128,541)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.67D4
C1C2C22C23C22×C4C23×C4C22×M4(2) — C24.67D4
C1C2C2×C4 — C24.67D4
C1C23C23×C4 — C24.67D4
C1C2C2C22×C4 — C24.67D4

Generators and relations for C24.67D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 316 in 158 conjugacy classes, 68 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C2×C4 [×22], C23, C23 [×2], C23 [×6], C22⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×2], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, C2.C42 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C22.4Q16 [×2], C23.7Q8 [×2], C2×C4.Q8, C2×C2.D8, C22×M4(2), C24.67D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C8⋊C22 [×2], C8.C22 [×2], C23.7Q8, M4(2)⋊C4 [×2], C8⋊D4 [×2], C82D4, C8.D4, C24.67D4

Smallest permutation representation of C24.67D4
On 64 points
Generators in S64
(2 6)(4 8)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)(34 38)(36 40)(42 46)(44 48)(49 53)(51 55)
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 24 47 16)(2 19 48 11)(3 22 41 14)(4 17 42 9)(5 20 43 12)(6 23 44 15)(7 18 45 10)(8 21 46 13)(25 40 57 53)(26 35 58 56)(27 38 59 51)(28 33 60 54)(29 36 61 49)(30 39 62 52)(31 34 63 55)(32 37 64 50)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim11111112222244
type++++++++-++-
imageC1C2C2C2C2C2C4D4D4Q8D4C4○D4C8⋊C22C8.C22
kernelC24.67D4C22.4Q16C23.7Q8C2×C4.Q8C2×C2.D8C22×M4(2)C2×M4(2)C2×C8C22×C4C22×C4C24C2×C4C22C22
# reps12211184121422

Matrix representation of C24.67D4 in GL8(𝔽17)

10000000
816000000
001600000
000160000
00001000
00000100
000000160
0000134016
,
160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
160000000
016000000
00420000
000130000
00000010
0000134115
00000100
000010013
,
131000000
04000000
001680000
00410000
0000116914
00001341115
00001700
00001042

G:=sub<GL(8,GF(17))| [1,8,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,13,0,0,0,0,0,1,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,13,0,1,0,0,0,0,0,4,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,13],[13,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,16,4,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,11,13,1,1,0,0,0,0,6,4,7,0,0,0,0,0,9,11,0,4,0,0,0,0,14,15,0,2] >;

C24.67D4 in GAP, Magma, Sage, TeX

C_2^4._{67}D_4
% in TeX

G:=Group("C2^4.67D4");
// GroupNames label

G:=SmallGroup(128,541);
// by ID

G=gap.SmallGroup(128,541);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,2019,248]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

׿
×
𝔽