Copied to
clipboard

G = C825C2order 128 = 27

5th semidirect product of C82 and C2 acting faithfully

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

Aliases: C825C2, C8.13D8, C42.656C23, C4.2(C2×D8), C4⋊Q164C2, C4.4(C4○D8), (C2×C8).223D4, C4.4D88C2, C84D4.6C2, C2.5(C84D4), C4⋊Q8.81C22, (C4×C8).369C22, C2.9(C8.12D4), C41D4.43C22, C22.57(C41D4), (C2×C4).713(C2×D4), SmallGroup(128,441)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C825C2
C1C2C22C2×C4C42C4×C8C82 — C825C2
C1C22C42 — C825C2
C1C22C42 — C825C2
C1C22C22C42 — C825C2

Generators and relations for C825C2
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a3b4, cbc=a4b3 >

Subgroups: 304 in 106 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×12], Q8 [×4], C23 [×2], C42, C4⋊C4 [×4], C2×C8 [×6], D8 [×4], Q16 [×4], C2×D4 [×6], C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4 [×8], C41D4 [×2], C4⋊Q8 [×2], C2×D8 [×2], C2×Q16 [×2], C82, C4.4D8 [×4], C84D4, C4⋊Q16, C825C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C4○D8 [×4], C84D4, C8.12D4 [×2], C825C2

Smallest permutation representation of C825C2
On 64 points
Generators in S64
(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 13 57 31 22 47 38 49)(2 14 58 32 23 48 39 50)(3 15 59 25 24 41 40 51)(4 16 60 26 17 42 33 52)(5 9 61 27 18 43 34 53)(6 10 62 28 19 44 35 54)(7 11 63 29 20 45 36 55)(8 12 64 30 21 46 37 56)
(2 17)(3 7)(4 23)(6 21)(8 19)(9 31)(10 52)(11 29)(12 50)(13 27)(14 56)(15 25)(16 54)(20 24)(26 44)(28 42)(30 48)(32 46)(33 39)(34 61)(35 37)(36 59)(38 57)(40 63)(41 51)(43 49)(45 55)(47 53)(58 60)(62 64)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H8A···8X
order1222224···4448···8
size111116162···216162···2

38 irreducible representations

dim11111222
type+++++++
imageC1C2C2C2C2D4D8C4○D8
kernelC825C2C82C4.4D8C84D4C4⋊Q16C2×C8C8C4
# reps114116816

Matrix representation of C825C2 in GL4(𝔽17) generated by

01300
4000
0033
00143
,
51200
5500
00314
0033
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,3,14,0,0,3,3],[5,5,0,0,12,5,0,0,0,0,3,3,0,0,14,3],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C825C2 in GAP, Magma, Sage, TeX

C_8^2\rtimes_5C_2
% in TeX

G:=Group("C8^2:5C2");
// GroupNames label

G:=SmallGroup(128,441);
// by ID

G=gap.SmallGroup(128,441);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,268,1123,136,2804,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^3*b^4,c*b*c=a^4*b^3>;
// generators/relations

׿
×
𝔽