Copied to
clipboard

G = C232D8order 128 = 27

1st semidirect product of C23 and D8 acting via D8/C4=C22

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

Aliases: C232D8, C24.79D4, C4⋊C44D4, (C2×C8)⋊5D4, (C2×D4)⋊5D4, (C22×D8)⋊1C2, C4.60C22≀C2, C22.80(C2×D8), C4.26(C41D4), C4.13(C4⋊D4), C2.12(C87D4), C2.12(C82D4), (C22×C4).137D4, C23.896(C2×D4), C2.27(D4⋊D4), C2.18(C22⋊D8), C22.4Q1636C2, C2.4(C232D4), C22.192C22≀C2, C22.100(C4○D8), (C22×C8).105C22, (C23×C4).267C22, (C22×D4).56C22, C22.217(C4⋊D4), C22.127(C8⋊C22), (C22×C4).1430C23, (C2×C4⋊D4)⋊1C2, (C2×C22⋊C8)⋊18C2, (C2×D4⋊C4)⋊10C2, (C2×C4).1020(C2×D4), (C2×C4).613(C4○D4), (C2×C4⋊C4).101C22, SmallGroup(128,731)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C232D8
C1C2C22C23C22×C4C23×C4C2×C4⋊D4 — C232D8
C1C2C22×C4 — C232D8
C1C23C23×C4 — C232D8
C1C2C2C22×C4 — C232D8

Generators and relations for C232D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 656 in 248 conjugacy classes, 56 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×5], C22 [×3], C22 [×4], C22 [×30], C8 [×3], C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×28], C23, C23 [×2], C23 [×22], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], D8 [×8], C22×C4 [×2], C22×C4 [×8], C2×D4 [×4], C2×D4 [×26], C24, C24 [×2], C22⋊C8 [×2], D4⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C22×C8 [×2], C2×D8 [×6], C23×C4, C22×D4 [×2], C22×D4 [×2], C22.4Q16, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C4⋊D4 [×2], C22×D8, C232D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, D8 [×2], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×D8, C4○D8, C8⋊C22 [×2], C232D4, C22⋊D8 [×2], D4⋊D4 [×2], C87D4, C82D4, C232D8

Smallest permutation representation of C232D8
On 64 points
Generators in S64
(1 26)(2 11)(3 28)(4 13)(5 30)(6 15)(7 32)(8 9)(10 33)(12 35)(14 37)(16 39)(17 51)(18 60)(19 53)(20 62)(21 55)(22 64)(23 49)(24 58)(25 40)(27 34)(29 36)(31 38)(41 57)(42 50)(43 59)(44 52)(45 61)(46 54)(47 63)(48 56)
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 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)
(1 8)(2 7)(3 6)(4 5)(9 45)(10 44)(11 43)(12 42)(13 41)(14 48)(15 47)(16 46)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 40)(34 39)(35 38)(36 37)(49 56)(50 55)(51 54)(52 53)(57 64)(58 63)(59 62)(60 61)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12···222222244444444448···8
size11···144888822224488884···4

32 irreducible representations

dim111111222222224
type+++++++++++++
imageC1C2C2C2C2C2D4D4D4D4D4C4○D4D8C4○D8C8⋊C22
kernelC232D8C22.4Q16C2×C22⋊C8C2×D4⋊C4C2×C4⋊D4C22×D8C4⋊C4C2×C8C22×C4C2×D4C24C2×C4C23C22C22
# reps111221421412442

Matrix representation of C232D8 in GL6(𝔽17)

1600000
0160000
000100
001000
000001
000010
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1430000
14140000
0016000
0001600
0000016
000010
,
1430000
330000
0016000
000100
0000016
0000160

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

C232D8 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_8
% in TeX

G:=Group("C2^3:2D8");
// GroupNames label

G:=SmallGroup(128,731);
// by ID

G=gap.SmallGroup(128,731);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,248]);
// Polycyclic

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

׿
×
𝔽