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G = C24.105D4order 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, D48(C2×D4), Q88(C2×D4), C4○D414D4, Q8⋊D42C2, C4⋊C4.8C23, (C2×C8).9C23, D4⋊D414C2, C22⋊D812C2, (C2×D8)⋊15C22, C22⋊C85C22, C4.43(C22×D4), C4.106C22≀C2, C4⋊D452C22, C225(C8⋊C22), C24.4C45C2, (C2×C4).225C24, (C2×SD16)⋊4C22, (C2×D4).28C23, C23.648(C2×D4), (C22×C4).787D4, D4⋊C410C22, Q8⋊C413C22, C22.17C22≀C2, C23.36D41C2, (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), (C22×C4○D4)⋊10C2, (C2×C4○D4)⋊64C22, SmallGroup(128,1738)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.105D4
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C24.105D4
C1C2C2×C4 — C24.105D4
C1C22C23×C4 — C24.105D4
C1C2C2C2×C4 — C24.105D4

Generators and relations for C24.105D4
 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 >

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

Smallest permutation representation of C24.105D4
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)])

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

Matrix representation of C24.105D4 in 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] >;

C24.105D4 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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