Copied to
clipboard

G = C42.23D4  order 128 = 27

5th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.23D4, (C4×C8).2C4, C8⋊C4.1C4, C22⋊C8.1C4, C42.31(C2×C4), (C2×C4).31C42, C4.1(C8.C4), (C22×C4).17Q8, C23.12(C4⋊C4), (C22×C4).119D4, C2.6(C4.9C42), C42.6C4.5C2, C42.12C4.8C2, C2.3(C4.C42), (C2×C42).127C22, C2.5(M4(2)⋊4C4), C22.37(C2.C42), (C2×C4).14(C4⋊C4), (C22×C4).91(C2×C4), (C2×C4).294(C22⋊C4), SmallGroup(128,19)

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

Derived series
C1 — C2×C4 — C42.23D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C42.12C4 — C42.23D4
Lower central
C1 — C2 — C2×C4 — C42.23D4
Upper central
C1 — C22 — C2×C42 — C42.23D4
Jennings
C1 — C22 — C22 — C2×C42 — C42.23D4

Generators and relations for C42.23D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a-1b, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a-1b2c3 >

Subgroups: 112 in 63 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C8⋊C4, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C42.12C4, C42.6C4, C42.23D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C8.C4, C4.9C42, C4.C42, M4(2)⋊4C4, C42.23D4

Smallest permutation representation of C42.23D4
►On 64 points
Generators in S64
(1 42 31 15)(2 43 32 16)(3 44 25 9)(4 45 26 10)(5 46 27 11)(6 47 28 12)(7 48 29 13)(8 41 30 14)(17 64 55 35)(18 57 56 36)(19 58 49 37)(20 59 50 38)(21 60 51 39)(22 61 52 40)(23 62 53 33)(24 63 54 34)

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

32 conjugacy classes

class 1 2A2B2C2D4A···4J4K8A···8H8I···8P
order122224···448···88···8
size111142···244···48···8

32 irreducible representations

dim111111222244
type+++++-
imageC1C2C2C4C4C4D4D4Q8C8.C4C4.9C42M4(2)⋊4C4
kernelC42.23D4C42.12C4C42.6C4C4×C8C8⋊C4C22⋊C8C42C22×C4C22×C4C4C2C2
# reps112444211822

Matrix representation of C42.23D4 ►in GL6(𝔽17)

100000
010000
001900
00131600
000019
00001316
,
1300000
0130000
0013000
0001300
000040
000004
,
1500000
580000
000010
000001
004000
000400
,
910000
080000
000084
000009
008000
002900

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,1,13,0,0,0,0,9,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[15,5,0,0,0,0,0,8,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,8,2,0,0,0,0,0,9,0,0,8,0,0,0,0,0,4,9,0,0] >;
 

C42.23D4 in GAP, Magma, Sage, TeX 

C_4^2._{23}D_4
 
% in TeX
 
G:=Group("C4^2.23D4");
 
// GroupNames label
 
G:=SmallGroup(128,19);
 
// by ID
 
G=gap.SmallGroup(128,19);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,136,3924,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^-1*b,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b^2*c^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁