Copied to
clipboard

G = C42.160D4  order 128 = 27

142nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.160D4, C23.208C24, C24.199C23, C22.312- (1+4), C22.472+ (1+4), C4.37(C4×D4), C42⋊18(C2×C4), C4.4D4⋊22C4, C42⋊4C4⋊11C2, C42⋊9C4⋊11C2, C23.8Q8⋊9C2, C22.99(C23×C4), C23.12(C22×C4), (C23×C4).47C22, C22.96(C22×D4), (C2×C42).415C22, C24.C22⋊8C2, (C22×C4).473C23, C2.6(C22.29C24), (C22×D4).479C22, (C22×Q8).399C22, C23.67C23⋊14C2, C24.3C22.25C2, C2.C42.44C22, C2.6(C22.36C24), C2.6(C23.38C23), C2.15(C23.33C23), (C2×C4×Q8)⋊5C2, C2.25(C2×C4×D4), (C2×C4×D4).32C2, (C2×Q8)⋊24(C2×C4), C22⋊C4⋊12(C2×C4), (C2×D4).168(C2×C4), (C2×C4).1188(C2×D4), (C2×C4).29(C22×C4), C22.93(C2×C4○D4), (C2×C4).648(C4○D4), (C2×C4⋊C4).180C22, (C2×C4.4D4).15C2, (C2×C22⋊C4).28C22, SmallGroup(128,1058)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C42.160D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊4C4 — C42.160D4
Lower central
C1 — C22 — C42.160D4
Upper central
C1 — C23 — C42.160D4
Jennings
C1 — C23 — C42.160D4

Subgroups: 540 in 306 conjugacy classes, 148 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×18], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×18], C2×C4 [×38], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×8], C42 [×6], C22⋊C4 [×16], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42 [×3], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×6], C4×D4 [×4], C4×Q8 [×4], C4.4D4 [×8], C23×C4 [×2], C22×D4, C22×Q8, C42⋊4C4, C42⋊9C4, C23.8Q8 [×4], C24.C22 [×4], C24.3C22, C23.67C23, C2×C4×D4, C2×C4×Q8, C2×C4.4D4, C42.160D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C2×C4×D4, C23.33C23 [×2], C22.29C24, C23.38C23, C22.36C24 [×2], C42.160D4

Generators and relations
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

20000000
02000000
00100000
00010000
00004010
00000401
00003010
00000301
,
40000000
04000000
00400000
00040000
00000100
00004000
00000001
00000040
,
01000000
40000000
00010000
00400000
00002030
00000203
00000030
00000003
,
30000000
02000000
00100000
00040000
00000104
00001040
00000204
00002040

G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,4,0,3,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,3,0,3],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,4,0,0,0,0,4,0,4,0] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4AF
order12···222224···44···4
size11···144442···24···4

44 irreducible representations

dim111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42+ (1+4)2- (1+4)
kernelC42.160D4C42⋊4C4C42⋊9C4C23.8Q8C24.C22C24.3C22C23.67C23C2×C4×D4C2×C4×Q8C2×C4.4D4C4.4D4C42C2×C4C22C22
# reps1114411111164422

In GAP, Magma, Sage, TeX 

C_4^2._{160}D_4
 
% in TeX
 
G:=Group("C4^2.160D4");
 
// GroupNames label
 
G:=SmallGroup(128,1058);
 
// by ID
 
G=gap.SmallGroup(128,1058);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀