Copied to
clipboard

G = C4.10D43C4order 128 = 27

2nd semidirect product of C4.10D4 and C4 acting via C4/C2=C2

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

Aliases: C4.71(C4×D4), C4⋊C4.215D4, (C2×C8).329D4, C4.10D43C4, M4(2).7(C2×C4), C2.3(D4.5D4), C2.4(D4.3D4), C4.28(C4.4D4), C4.C42.2C2, C23.265(C4○D4), C82M4(2).18C2, M4(2)⋊C4.2C2, (C22×C4).693C23, (C22×C8).395C22, C23.38D4.5C2, (C22×Q8).29C22, C22.128(C4⋊D4), C22.C42.11C2, C22.6(C422C2), C22.9(C42⋊C2), C4.90(C22.D4), C42⋊C2.273C22, (C2×M4(2)).198C22, C2.15(C24.C22), (C2×Q8).82(C2×C4), (C2×C4).1338(C2×D4), (C2×C4⋊C4).68C22, (C2×C4).15(C22×C4), (C2×C4).334(C4○D4), (C2×Q8⋊C4).32C2, (C2×C4.10D4).2C2, SmallGroup(128,662)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.10D43C4
C1C2C22C23C22×C4C2×M4(2)C82M4(2) — C4.10D43C4
C1C2C2×C4 — C4.10D43C4
C1C22C22×C4 — C4.10D43C4
C1C2C2C22×C4 — C4.10D43C4

Generators and relations for C4.10D43C4
 G = < a,b,c,d | a4=d4=1, b4=a2, c2=bab-1=dad-1=a-1, dcd-1=ac=ca, cbc-1=a-1b3, dbd-1=ab3 >

Subgroups: 196 in 106 conjugacy classes, 48 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×5], C22 [×3], C22 [×2], C8 [×7], C2×C4 [×6], C2×C4 [×8], Q8 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×5], C4×C8, C8⋊C4, C4.10D4 [×4], Q8⋊C4 [×4], C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2) [×3], C22×Q8, C4.C42, C22.C42, C82M4(2), C2×C4.10D4, C2×Q8⋊C4, C23.38D4, M4(2)⋊C4, C4.10D43C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C24.C22, D4.3D4, D4.5D4, C4.10D43C4

Smallest permutation representation of C4.10D43C4
On 64 points
Generators in S64
(1 21 5 17)(2 18 6 22)(3 23 7 19)(4 20 8 24)(9 64 13 60)(10 61 14 57)(11 58 15 62)(12 63 16 59)(25 40 29 36)(26 37 30 33)(27 34 31 38)(28 39 32 35)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(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 57 17 14 5 61 21 10)(2 13 22 64 6 9 18 60)(3 63 19 12 7 59 23 16)(4 11 24 62 8 15 20 58)(25 53 36 48 29 49 40 44)(26 47 33 52 30 43 37 56)(27 51 38 46 31 55 34 42)(28 45 35 50 32 41 39 54)
(1 51 19 48)(2 45 20 56)(3 49 21 46)(4 43 22 54)(5 55 23 44)(6 41 24 52)(7 53 17 42)(8 47 18 50)(9 32 62 33)(10 38 63 29)(11 30 64 39)(12 36 57 27)(13 28 58 37)(14 34 59 25)(15 26 60 35)(16 40 61 31)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E···8J8K8L8M8N
order12222244444444444488888···88888
size11112222224444888822224···48888

32 irreducible representations

dim111111111222244
type++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4C4○D4C4○D4D4.3D4D4.5D4
kernelC4.10D43C4C4.C42C22.C42C82M4(2)C2×C4.10D4C2×Q8⋊C4C23.38D4M4(2)⋊C4C4.10D4C4⋊C4C2×C8C2×C4C23C2C2
# reps111111118226222

Matrix representation of C4.10D43C4 in GL6(𝔽17)

1600000
0160000
00161600
002100
000011
00001516
,
400000
040000
000010
000001
00161600
002100
,
0160000
100000
000005
0000100
0010500
007700
,
0130000
1300000
001000
00151600
000010
00001516

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,1,15,0,0,0,0,1,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,10,7,0,0,0,0,5,7,0,0,0,10,0,0,0,0,5,0,0,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,1,15,0,0,0,0,0,16,0,0,0,0,0,0,1,15,0,0,0,0,0,16] >;

C4.10D43C4 in GAP, Magma, Sage, TeX

C_4._{10}D_4\rtimes_3C_4
% in TeX

G:=Group("C4.10D4:3C4");
// GroupNames label

G:=SmallGroup(128,662);
// by ID

G=gap.SmallGroup(128,662);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,718]);
// Polycyclic

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

׿
×
𝔽