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G = C24.104D4  order 128 = 27

59th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.104D4, (C2×D4)⋊50D4, (C2×Q8)⋊37D4, D4.41(C2×D4), C4⋊C4.7C23, (C2×C8).8C23, Q8.41(C2×D4), D4⋊D4⋊13C2, C4.83C22≀C2, (C2×D8)⋊14C22, C22⋊C8⋊4C22, C4.42(C22×D4), D4.7D4⋊13C2, D4⋊C4⋊9C22, C4⋊D4⋊51C22, C24.4C4⋊4C2, (C2×C4).224C24, (C2×Q16)⋊13C22, (C2×SD16)⋊3C22, (C2×D4).27C23, (C22×C4).715D4, C23.230(C2×D4), C22⋊Q8⋊63C22, (C2×Q8).20C23, C22.19C24⋊4C2, Q8⋊C4⋊12C22, C22.59C22≀C2, C23.38D4⋊3C2, C23.37D4⋊3C2, C2.7(D8⋊C22), (C22×C4).962C23, (C23×C4).544C22, C22.484(C22×D4), C42⋊C2.96C22, (C22×D4).564C22, (C2×M4(2)).40C22, (C22×Q8).468C22, (C2×C8⋊C22)⋊6C2, (C22×C4○D4)⋊9C2, (C2×C8.C22)⋊6C2, C2.42(C2×C22≀C2), (C2×C4).1213(C2×D4), (C2×C4○D4).100C22, SmallGroup(128,1737)

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

Derived series
C1 — C2×C4 — C24.104D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C24.104D4
Lower central
C1 — C2 — C2×C4 — C24.104D4
Upper central
C1 — C22 — C23×C4 — C24.104D4
Jennings
C1 — C2 — C2 — C2×C4 — C24.104D4

Subgroups: 740 in 379 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×9], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×31], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×37], D4 [×4], D4 [×30], Q8 [×4], Q8 [×8], C23, C23 [×2], C23 [×15], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×18], C2×D4, C2×D4 [×6], C2×D4 [×18], C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×36], C24, C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C8⋊C22 [×4], C8.C22 [×4], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4 [×4], C2×C4○D4 [×10], C24.4C4, C23.37D4, C23.38D4, D4⋊D4 [×4], D4.7D4 [×4], C22.19C24, C2×C8⋊C22, C2×C8.C22, C22×C4○D4, C24.104D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D8⋊C22 [×2], C24.104D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=ac=ca, ad=da, faf=acd, bc=cb, ebe-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Smallest permutation representation
►On 32 points
Generators in S32
(2 25)(4 27)(6 29)(8 31)(9 20)(10 14)(11 22)(12 16)(13 24)(15 18)(17 21)(19 23)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

100000
0160000
001001
00016160
000010
0000016
,
100000
010000
00160016
00016160
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
0160000
100000
0040010
0001360
000840
0090013
,
0160000
1600000
000100
001000
00150016
00015160

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,1,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,16,0,0,1],[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],[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],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,9,0,0,0,13,8,0,0,0,0,6,4,0,0,0,10,0,0,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,15,0,0,0,1,0,0,15,0,0,0,0,0,16,0,0,0,0,16,0] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E2F···2K2L4A···4H4I4J4K4L4M4N4O8A8B8C8D
order1222222···224···444444448888
size1111224···482···244448888888

32 irreducible representations

dim111111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4D4D8⋊C22
kernelC24.104D4C24.4C4C23.37D4C23.38D4D4⋊D4D4.7D4C22.19C24C2×C8⋊C22C2×C8.C22C22×C4○D4C22×C4C2×D4C2×Q8C24C2
# reps111144111134414

In GAP, Magma, Sage, TeX 

C_2^4._{104}D_4
 
% in TeX
 
G:=Group("C2^4.104D4");
 
// GroupNames label
 
G:=SmallGroup(128,1737);
 
// by ID
 
G=gap.SmallGroup(128,1737);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2019,2804,1411,172]);
 
// 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=a*c=c*a,a*d=d*a,f*a*f=a*c*d,b*c=c*b,e*b*e^-1=f*b*f=b*d=d*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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