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G = C24.100D4order 128 = 27

55th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.100D4, C4.5(C22×Q8), (C2×M4(2))⋊8C4, C4.49(C23×C4), C8.13(C22×C4), C2.D863C22, C4.Q843C22, (C22×C4).65Q8, C4⋊C4.350C23, C23.32(C4⋊C4), M4(2)⋊15(C2×C4), (C2×C8).244C23, (C2×C4).187C24, C23.379(C2×D4), (C22×C4).785D4, C4(M4(2)⋊C4), M4(2)⋊C445C2, C2.2(D8⋊C22), C23.25D420C2, (C22×C8).243C22, (C23×C4).519C22, (C22×C4).906C23, C22.134(C22×D4), (C22×M4(2)).4C2, C42⋊C2.287C22, (C2×M4(2)).257C22, (C2×C8)⋊8(C2×C4), C4.66(C2×C4⋊C4), (C2×C4).62(C4⋊C4), C22.37(C2×C4⋊C4), C2.26(C22×C4⋊C4), (C2×C4).142(C2×Q8), (C2×C4).1568(C2×D4), (C2×C4⋊C4).905C22, (C2×C4).250(C22×C4), (C22×C4).331(C2×C4), (C2×C42⋊C2).55C2, SmallGroup(128,1643)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C24.100D4
C1C2C22C2×C4C22×C4C23×C4C22×M4(2) — C24.100D4
C1C2C4 — C24.100D4
C1C2×C4C23×C4 — C24.100D4
C1C2C2C2×C4 — C24.100D4

Generators and relations for C24.100D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 364 in 246 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×10], C8 [×8], C2×C4 [×2], C2×C4 [×26], C2×C4 [×16], C23, C23 [×6], C23 [×2], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], M4(2) [×16], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, C4.Q8 [×8], C2.D8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×8], C42⋊C2 [×4], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C23.25D4 [×4], M4(2)⋊C4 [×8], C2×C42⋊C2 [×2], C22×M4(2), C24.100D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, D8⋊C22 [×2], C24.100D4

Smallest permutation representation of C24.100D4
On 32 points
Generators in S32
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)
(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 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 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 14 23 29)(2 9 24 32)(3 12 17 27)(4 15 18 30)(5 10 19 25)(6 13 20 28)(7 16 21 31)(8 11 22 26)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4Z8A···8H
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim1111112224
type++++++-+
imageC1C2C2C2C2C4D4Q8D4D8⋊C22
kernelC24.100D4C23.25D4M4(2)⋊C4C2×C42⋊C2C22×M4(2)C2×M4(2)C22×C4C22×C4C24C2
# reps14821163414

Matrix representation of C24.100D4 in GL6(𝔽17)

1600000
0160000
001000
00161600
000010
00160016
,
100000
010000
001000
000100
0010160
00160016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1300000
940000
00161500
0011100
00616013
0011160
,
4130000
0130000
004008
00701613
0011104
0000013

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[13,9,0,0,0,0,0,4,0,0,0,0,0,0,16,11,6,1,0,0,15,1,16,1,0,0,0,0,0,16,0,0,0,0,13,0],[4,0,0,0,0,0,13,13,0,0,0,0,0,0,4,7,11,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,8,13,4,13] >;

C24.100D4 in GAP, Magma, Sage, TeX

C_2^4._{100}D_4
% in TeX

G:=Group("C2^4.100D4");
// GroupNames label

G:=SmallGroup(128,1643);
// by ID

G=gap.SmallGroup(128,1643);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,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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