Copied to
clipboard

G = C24.89D4order 128 = 27

44th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.89D4, C23.19SD16, (C2×C8).164D4, C2.22(C88D4), C2.22(C8⋊D4), C23.932(C2×D4), (C22×C4).162D4, C22.4Q1631C2, C4.56(C4.4D4), C4.20(C422C2), C22.124(C4○D8), (C23×C4).277C22, (C22×C8).325C22, C22.105(C2×SD16), C23.7Q8.22C2, C22.253(C4⋊D4), C22.152(C8⋊C22), (C22×C4).1466C23, C2.8(C23.46D4), C2.8(C23.47D4), C4.112(C22.D4), C2.12(C23.19D4), C2.12(C23.20D4), C2.12(C23.11D4), C22.141(C8.C22), C22.122(C22.D4), (C2×C4.Q8)⋊23C2, (C2×C4).1375(C2×D4), (C2×C22⋊C8).38C2, (C2×C4).628(C4○D4), (C2×C4⋊C4).151C22, SmallGroup(128,809)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.89D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C24.89D4
C1C2C22×C4 — C24.89D4
C1C23C23×C4 — C24.89D4
C1C2C2C22×C4 — C24.89D4

Generators and relations for C24.89D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=db=bd, eae-1=ab=ba, ac=ca, ad=da, faf-1=abc, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 288 in 128 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×5], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×19], C23, C23 [×2], C23 [×6], C22⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×10], C24, C2.C42 [×2], C22⋊C8 [×2], C4.Q8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C23×C4, C22.4Q16 [×3], C23.7Q8 [×2], C2×C22⋊C8, C2×C4.Q8, C24.89D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C2×SD16, C4○D8, C8⋊C22, C8.C22, C23.11D4, C88D4, C8⋊D4, C23.46D4, C23.19D4, C23.47D4, C23.20D4, C24.89D4

Smallest permutation representation of C24.89D4
On 64 points
Generators in S64
(2 12)(4 14)(6 16)(8 10)(17 34)(18 62)(19 36)(20 64)(21 38)(22 58)(23 40)(24 60)(26 50)(28 52)(30 54)(32 56)(33 45)(35 47)(37 41)(39 43)(42 57)(44 59)(46 61)(48 63)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
(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 22 15 47)(2 17 16 42)(3 20 9 45)(4 23 10 48)(5 18 11 43)(6 21 12 46)(7 24 13 41)(8 19 14 44)(25 60 53 37)(26 63 54 40)(27 58 55 35)(28 61 56 38)(29 64 49 33)(30 59 50 36)(31 62 51 39)(32 57 52 34)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111122222244
type+++++++++-
imageC1C2C2C2C2D4D4D4C4○D4SD16C4○D8C8⋊C22C8.C22
kernelC24.89D4C22.4Q16C23.7Q8C2×C22⋊C8C2×C4.Q8C2×C8C22×C4C24C2×C4C23C22C22C22
# reps13211211104411

Matrix representation of C24.89D4 in GL6(𝔽17)

100000
4160000
001000
0011600
000010
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1320000
040000
001000
000100
000020
000008
,
180000
4160000
0011500
0001600
000008
000020

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

C24.89D4 in GAP, Magma, Sage, TeX

C_2^4._{89}D_4
% in TeX

G:=Group("C2^4.89D4");
// GroupNames label

G:=SmallGroup(128,809);
// by ID

G=gap.SmallGroup(128,809);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,422,387,58,718,172]);
// Polycyclic

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