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G = C24.110D4order 128 = 27

65th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.110D4, (C2×C8)⋊10D4, C4(C82D4), C4(C8⋊D4), C8.21(C2×D4), C82D435C2, C8⋊D457C2, C4(C8.D4), C8.D435C2, (C2×D8)⋊47C22, C4⋊C4.23C23, C4.Q850C22, C2.D869C22, (C2×C4).258C24, (C2×C8).250C23, (C2×Q16)⋊52C22, (C2×D4).61C23, C4.152(C22×D4), C23.238(C2×D4), (C22×C4).428D4, (C2×Q8).49C23, C4.212(C4⋊D4), D4⋊C492C22, C22.19C247C2, Q8⋊C496C22, (C2×SD16)⋊55C22, (C22×M4(2))⋊3C2, C4⋊D4.149C22, C23.24D439C2, C23.25D426C2, C22.35(C4⋊D4), (C23×C4).550C22, (C22×C8).258C22, C22.518(C22×D4), C22⋊Q8.154C22, C2.14(D8⋊C22), (C22×C4).1537C23, C42⋊C2.107C22, (C2×M4(2)).263C22, (C2×C4○D8)⋊16C2, (C2×C4)(C8⋊D4), (C2×C4)(C82D4), C4.25(C2×C4○D4), (C2×C4)(C8.D4), (C2×C4).474(C2×D4), C2.76(C2×C4⋊D4), (C2×C4).704(C4○D4), (C2×C4○D4).124C22, SmallGroup(128,1786)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.110D4
C1C2C22C2×C4C22×C4C23×C4C22×M4(2) — C24.110D4
C1C2C2×C4 — C24.110D4
C1C2×C4C23×C4 — C24.110D4
C1C2C2C2×C4 — C24.110D4

Generators and relations for C24.110D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 484 in 250 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], M4(2) [×8], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C23×C4, C2×C4○D4 [×2], C23.24D4 [×2], C23.25D4, C8⋊D4 [×4], C82D4 [×2], C8.D4 [×2], C22.19C24 [×2], C22×M4(2), C2×C4○D8, C24.110D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D8⋊C22 [×2], C24.110D4

Smallest permutation representation of C24.110D4
On 32 points
Generators in S32
(2 6)(4 8)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(18 22)(20 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(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)
(1 29 5 25)(2 32 6 28)(3 27 7 31)(4 30 8 26)(9 24 13 20)(10 19 14 23)(11 22 15 18)(12 17 16 21)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order1222222222444444444···48···8
size1111224488111122448···84···4

32 irreducible representations

dim11111111122224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4D8⋊C22
kernelC24.110D4C23.24D4C23.25D4C8⋊D4C82D4C8.D4C22.19C24C22×M4(2)C2×C4○D8C2×C8C22×C4C24C2×C4C2
# reps12142221143144

Matrix representation of C24.110D4 in GL6(𝔽17)

100000
0160000
001000
0001600
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
000100
004000
000004
0000160
,
010000
100000
000004
0000160
000100
004000

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,4,0,0,0] >;

C24.110D4 in GAP, Magma, Sage, TeX

C_2^4._{110}D_4
% in TeX

G:=Group("C2^4.110D4");
// GroupNames label

G:=SmallGroup(128,1786);
// by ID

G=gap.SmallGroup(128,1786);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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^3>;
// generators/relations

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