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G = C24.100D4  order 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), C4.Q8⋊43C22, C2.D8⋊63C22, (C22×C4).65Q8, C4⋊C4.350C23, C23.32(C4⋊C4), M4(2)⋊15(C2×C4), (C2×C4).187C24, (C2×C8).244C23, (C22×C4).785D4, C23.379(C2×D4), C4○(M4(2)⋊C4), M4(2)⋊C4⋊45C2, C23.25D4⋊20C2, C2.2(D8⋊C22), (C22×C4).906C23, (C22×C8).243C22, (C23×C4).519C22, 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), C2.26(C22×C4⋊C4), C22.37(C2×C4⋊C4), (C2×C4).142(C2×Q8), (C2×C4).1568(C2×D4), (C2×C4⋊C4).905C22, (C22×C4).331(C2×C4), (C2×C4).250(C22×C4), (C2×C42⋊C2).55C2, SmallGroup(128,1643)

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

Derived series
C1 — C4 — C24.100D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.100D4
Lower central
C1 — C2 — C4 — C24.100D4
Upper central
C1 — C2×C4 — C23×C4 — C24.100D4
Jennings
C1 — C2 — C2 — C2×C4 — C24.100D4

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

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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

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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