Copied to
clipboard

G = C24.95D4  order 128 = 27

50th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.95D4, C25.37C22, C23.312C24, C24.562C23, C22.1282+ (1+4), C22⋊C4⋊33D4, C23.152(C2×D4), C2.18(D4⋊5D4), C23.10D4⋊9C2, C23.11D4⋊3C2, C2.9(C23⋊3D4), (C22×C4).48C23, C23.7Q8⋊33C2, C23.8Q8⋊28C2, C23.299(C4○D4), C23.23D4⋊27C2, C23.34D4⋊19C2, C22.73(C4⋊D4), (C23×C4).330C22, C22.192(C22×D4), C2.C42⋊20C22, (C22×D4).118C22, C22⋊1(C22.D4), C2.11(C22.45C24), (C2×C4⋊C4)⋊13C22, (C2×C4).306(C2×D4), C2.16(C2×C4⋊D4), (C2×C22≀C2).8C2, (C2×C22⋊C4)⋊14C22, (C22×C22⋊C4)⋊19C2, C22.191(C2×C4○D4), (C2×C22.D4)⋊5C2, C2.14(C2×C22.D4), SmallGroup(128,1144)

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

Derived series
C1 — C23 — C24.95D4
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C24.95D4
Lower central
C1 — C23 — C24.95D4
Upper central
C1 — C23 — C24.95D4
Jennings
C1 — C23 — C24.95D4

Subgroups: 900 in 407 conjugacy classes, 116 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×11], C4 [×13], C22 [×3], C22 [×12], C22 [×57], C2×C4 [×4], C2×C4 [×47], D4 [×12], C23, C23 [×14], C23 [×57], C22⋊C4 [×4], C22⋊C4 [×26], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×8], C22×C4 [×20], C2×D4 [×16], C24 [×4], C24 [×4], C24 [×10], C2.C42 [×8], C2×C22⋊C4 [×5], C2×C22⋊C4 [×10], C2×C22⋊C4 [×8], C2×C4⋊C4 [×4], C22≀C2 [×4], C22.D4 [×4], C23×C4 [×2], C23×C4 [×2], C22×D4 [×3], C25, C23.7Q8 [×2], C23.34D4, C23.8Q8, C23.23D4, C23.23D4 [×2], C23.10D4 [×2], C23.11D4 [×2], C22×C22⋊C4 [×2], C2×C22≀C2, C2×C22.D4, C24.95D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C4⋊D4, C2×C22.D4, C23⋊3D4, D4⋊5D4 [×2], C22.45C24 [×2], C24.95D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=c, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Smallest permutation representation
►On 32 points
Generators in S32
(2 24)(4 22)(5 20)(6 30)(7 18)(8 32)(9 29)(10 17)(11 31)(12 19)(14 26)(16 28)

(1 27)(2 16)(3 25)(4 14)(5 20)(6 30)(7 18)(8 32)(9 29)(10 17)(11 31)(12 19)(13 21)(15 23)(22 26)(24 28)

(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)

(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

(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)

(1 6 23 10)(2 5 24 9)(3 8 21 12)(4 7 22 11)(13 32 25 19)(14 31 26 18)(15 30 27 17)(16 29 28 20)
 
G:=sub<Sym(32)| (2,24)(4,22)(5,20)(6,30)(7,18)(8,32)(9,29)(10,17)(11,31)(12,19)(14,26)(16,28), (1,27)(2,16)(3,25)(4,14)(5,20)(6,30)(7,18)(8,32)(9,29)(10,17)(11,31)(12,19)(13,21)(15,23)(22,26)(24,28), (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24), (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), (1,6,23,10)(2,5,24,9)(3,8,21,12)(4,7,22,11)(13,32,25,19)(14,31,26,18)(15,30,27,17)(16,29,28,20)>;
 
G:=Group( (2,24)(4,22)(5,20)(6,30)(7,18)(8,32)(9,29)(10,17)(11,31)(12,19)(14,26)(16,28), (1,27)(2,16)(3,25)(4,14)(5,20)(6,30)(7,18)(8,32)(9,29)(10,17)(11,31)(12,19)(13,21)(15,23)(22,26)(24,28), (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24), (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), (1,6,23,10)(2,5,24,9)(3,8,21,12)(4,7,22,11)(13,32,25,19)(14,31,26,18)(15,30,27,17)(16,29,28,20) );
 
G=PermutationGroup([(2,24),(4,22),(5,20),(6,30),(7,18),(8,32),(9,29),(10,17),(11,31),(12,19),(14,26),(16,28)], [(1,27),(2,16),(3,25),(4,14),(5,20),(6,30),(7,18),(8,32),(9,29),(10,17),(11,31),(12,19),(13,21),(15,23),(22,26),(24,28)], [(1,23),(2,24),(3,21),(4,22),(5,9),(6,10),(7,11),(8,12),(13,25),(14,26),(15,27),(16,28),(17,30),(18,31),(19,32),(20,29)], [(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24)], [(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)], [(1,6,23,10),(2,5,24,9),(3,8,21,12),(4,7,22,11),(13,32,25,19),(14,31,26,18),(15,30,27,17),(16,29,28,20)])
 

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000400
000010
000004
,
100000
010000
001000
000400
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
400000
000100
004000
000010
000001
,
040000
400000
000200
002000
000001
000010

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

38 conjugacy classes

class 1 2A···2G2H···2O2P2Q2R4A···4P4Q4R4S
order12···22···22224···4444
size11···12···24484···4888

38 irreducible representations

dim11111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)
kernelC24.95D4C23.7Q8C23.34D4C23.8Q8C23.23D4C23.10D4C23.11D4C22×C22⋊C4C2×C22≀C2C2×C22.D4C22⋊C4C24C23C22
# reps121132221144122

In GAP, Magma, Sage, TeX 

C_2^4._{95}D_4
 
% in TeX
 
G:=Group("C2^4.95D4");
 
// GroupNames label
 
G:=SmallGroup(128,1144);
 
// by ID
 
G=gap.SmallGroup(128,1144);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,100,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀