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G = C24.61D4order 128 = 27

16th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.61D4, C23.7Q16, C23.16SD16, C22⋊Q88C4, (C22×Q8)⋊5C4, C4.14(C23⋊C4), C23.510(C2×D4), (C22×C4).221D4, C22.10(C2×Q16), C23.31D42C2, C22.11(C2×SD16), C23.7Q8.4C2, C22⋊C8.168C22, (C22×C4).642C23, (C23×C4).214C22, C22⋊Q8.147C22, C23.176(C22⋊C4), C22.21(Q8⋊C4), C2.C42.9C22, C2.25(C42⋊C22), (C2×C4⋊C4)⋊12C4, C4⋊C4.20(C2×C4), C2.23(C2×C23⋊C4), (C2×Q8).15(C2×C4), (C2×C22⋊Q8).5C2, (C2×C4).1166(C2×D4), (C2×C22⋊C8).11C2, C2.10(C2×Q8⋊C4), (C2×C4).132(C22×C4), (C22×C4).205(C2×C4), (C2×C4).242(C22⋊C4), C22.196(C2×C22⋊C4), SmallGroup(128,252)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.61D4
C1C2C22C23C22×C4C23×C4C2×C22⋊Q8 — C24.61D4
C1C22C2×C4 — C24.61D4
C1C22C23×C4 — C24.61D4
C1C2C22C22×C4 — C24.61D4

Generators and relations for C24.61D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, faf-1=acd, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 324 in 144 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C8 [×2], C2×C4 [×4], C2×C4 [×23], Q8 [×4], C23 [×3], C23 [×4], C23 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×6], C22×C4 [×6], C2×Q8 [×2], C2×Q8 [×3], C24, C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C23×C4, C22×Q8, C23.31D4 [×4], C23.7Q8, C2×C22⋊C8, C2×C22⋊Q8, C24.61D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C23⋊C4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C2×C23⋊C4, C2×Q8⋊C4, C42⋊C22, C24.61D4

Smallest permutation representation of C24.61D4
On 32 points
Generators in S32
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(2 31)(4 25)(6 27)(8 29)(10 20)(12 22)(14 24)(16 18)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 17)(2 20 31 10)(3 13)(4 16 25 18)(5 21)(6 24 27 14)(7 9)(8 12 29 22)(11 26)(15 30)(19 28)(23 32)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E4F4G···4N8A···8H
order12222···24444444···48···8
size11112···22222448···84···4

32 irreducible representations

dim11111111222244
type+++++++-+
imageC1C2C2C2C2C4C4C4D4D4SD16Q16C23⋊C4C42⋊C22
kernelC24.61D4C23.31D4C23.7Q8C2×C22⋊C8C2×C22⋊Q8C2×C4⋊C4C22⋊Q8C22×Q8C22×C4C24C23C23C4C2
# reps14111242314422

Matrix representation of C24.61D4 in GL6(𝔽17)

100000
010000
000400
0013000
0000013
000040
,
1600000
0160000
001000
000100
00150160
0002016
,
1600000
0160000
001000
000100
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
3140000
330000
000809
009090
000108
001090
,
040000
400000
000100
001000
0003013
00140130

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,15,0,0,0,0,1,0,2,0,0,0,0,16,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,9,0,1,0,0,8,0,1,0,0,0,0,9,0,9,0,0,9,0,8,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,14,0,0,1,0,3,0,0,0,0,0,0,13,0,0,0,0,13,0] >;

C24.61D4 in GAP, Magma, Sage, TeX

C_2^4._{61}D_4
% in TeX

G:=Group("C2^4.61D4");
// GroupNames label

G:=SmallGroup(128,252);
// by ID

G=gap.SmallGroup(128,252);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,387,352,1123,1018,248,1971]);
// Polycyclic

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

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