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G = C24.105D4  order 128 = 27

60th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.105D4, D4⋊8(C2×D4), Q8⋊8(C2×D4), C4○D4⋊14D4, Q8⋊D4⋊2C2, C4⋊C4.8C23, (C2×C8).9C23, C22⋊D8⋊12C2, D4⋊D4⋊14C2, (C2×D8)⋊15C22, C22⋊C8⋊5C22, C4.43(C22×D4), C4.106C22≀C2, C4⋊D4⋊52C22, C22⋊5(C8⋊C22), (C2×C4).225C24, C24.4C4⋊5C2, (C2×SD16)⋊4C22, (C2×D4).28C23, C23.648(C2×D4), (C22×C4).787D4, D4⋊C4⋊10C22, Q8⋊C4⋊13C22, C22.17C22≀C2, C23.36D4⋊1C2, (C22×D4)⋊17C22, (C2×M4(2))⋊2C22, (C2×Q8).353C23, (C22×Q8)⋊54C22, C2.8(D8⋊C22), (C23×C4).545C22, (C22×C4).963C23, C22.485(C22×D4), (C2×C8⋊C22)⋊7C2, (C2×C4⋊D4)⋊45C2, (C2×C4⋊C4)⋊46C22, (C2×C4).452(C2×D4), C2.11(C2×C8⋊C22), C2.43(C2×C22≀C2), (C2×C4○D4)⋊64C22, (C22×C4○D4)⋊10C2, SmallGroup(128,1738)

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

Derived series
C1 — C2×C4 — C24.105D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C24.105D4
Lower central
C1 — C2 — C2×C4 — C24.105D4
Upper central
C1 — C22 — C23×C4 — C24.105D4
Jennings
C1 — C2 — C2 — C2×C4 — C24.105D4

Subgroups: 844 in 403 conjugacy classes, 110 normal (28 characteristic)
C1, C2 [×3], C2 [×11], C4 [×4], C4 [×8], C22, C22 [×4], C22 [×37], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×36], D4 [×4], D4 [×36], Q8 [×4], Q8 [×6], C23 [×3], C23 [×23], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4 [×6], C22×C4 [×18], C2×D4 [×4], C2×D4 [×30], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×8], C4○D4 [×28], C24, C24 [×2], C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×4], C8⋊C22 [×8], C23×C4, C23×C4, C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C4○D4 [×4], C2×C4○D4 [×10], C24.4C4, C23.36D4 [×2], C22⋊D8 [×2], Q8⋊D4 [×2], D4⋊D4 [×4], C2×C4⋊D4, C2×C8⋊C22 [×2], C22×C4○D4, C24.105D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C8⋊C22 [×2], C22×D4 [×3], C2×C22≀C2, C2×C8⋊C22, D8⋊C22, C24.105D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Smallest permutation representation
►On 32 points
Generators in S32
(1 22)(2 13)(3 24)(4 15)(5 18)(6 9)(7 20)(8 11)(10 27)(12 29)(14 31)(16 25)(17 32)(19 26)(21 28)(23 30)

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

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

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

(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)

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

Matrix representation ►G ⊆ GL6(ℤ)

1-10000
0-10000
00-1000
000-100
0000-10
00000-1
,
100000
010000
001000
000100
0000-10
00-110-1
,
-100000
0-10000
001000
000100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
100000
2-10000
00-111-2
000010
001000
001001
,
-100000
-210000
001000
000-100
00-111-2
00-100-1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,-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,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-1,0,0,0,1,0,1,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[1,2,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,-2,0,0,1],[-1,-2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,-2,-1] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2L2M2N4A···4F4G···4K4L4M8A8B8C8D
order122222222···2224···44···4448888
size111122224···4882···24···4888888

32 irreducible representations

dim11111111122244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4C8⋊C22D8⋊C22
kernelC24.105D4C24.4C4C23.36D4C22⋊D8Q8⋊D4D4⋊D4C2×C4⋊D4C2×C8⋊C22C22×C4○D4C22×C4C4○D4C24C22C2
# reps11222412138122

In GAP, Magma, Sage, TeX 

C_2^4._{105}D_4
 
% in TeX
 
G:=Group("C2^4.105D4");
 
// GroupNames label
 
G:=SmallGroup(128,1738);
 
// by ID
 
G=gap.SmallGroup(128,1738);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,248,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^3>;
 
// generators/relations
 

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