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G = C24.98D4  order 128 = 27

53rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.98D4, C4.7(C23×C4), C4⋊C4.343C23, (C2×C4).177C24, (C2×C8).392C23, (C22×C8)⋊48C22, D4.19(C22×C4), C4.142(C22×D4), C23.377(C2×D4), (C22×C4).781D4, Q8.19(C22×C4), D4⋊C4⋊85C22, Q8⋊C4⋊88C22, (C2×D4).361C23, C4○(C23.36D4), C4○(C23.38D4), C4○(C23.37D4), (C2×Q8).334C23, C42⋊C2⋊75C22, C23.24D4⋊34C2, C23.37D4⋊38C2, C23.36D4⋊47C2, C2.1(D8⋊C22), C23.38D4⋊38C2, C23.87(C22⋊C4), (C2×M4(2))⋊70C22, (C22×M4(2))⋊20C2, (C22×C4).901C23, (C23×C4).515C22, C22.127(C22×D4), (C22×D4).554C22, (C22×Q8).458C22, (C2×C4○D4)⋊20C4, C4○D4⋊14(C2×C4), (C2×D4)⋊49(C2×C4), (C2×Q8)⋊40(C2×C4), (C2×C4).445(C2×D4), C4.75(C2×C22⋊C4), (C2×C4⋊C4)⋊115C22, (C2×C42⋊C2)⋊42C2, (C2×C4).245(C22×C4), (C22×C4).325(C2×C4), (C22×C4○D4).20C2, C22.22(C2×C22⋊C4), C2.39(C22×C22⋊C4), (C2×C4).159(C22⋊C4), (C2×C4○D4).274C22, (C2×C4)○(C23.38D4), (C2×C4)○(C23.36D4), SmallGroup(128,1628)

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

Derived series
C1 — C4 — C24.98D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C24.98D4
Lower central
C1 — C2 — C4 — C24.98D4
Upper central
C1 — C2×C4 — C23×C4 — C24.98D4
Jennings
C1 — C2 — C2 — C2×C4 — C24.98D4

Subgroups: 668 in 386 conjugacy classes, 172 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×26], C8 [×4], C2×C4 [×4], C2×C4 [×24], C2×C4 [×30], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23 [×3], C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×6], C22×C4 [×8], C22×C4 [×15], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24, D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C23.24D4 [×4], C23.36D4 [×4], C23.37D4 [×2], C23.38D4 [×2], C2×C42⋊C2, C22×M4(2), C22×C4○D4, C24.98D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, D8⋊C22 [×2], C24.98D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Smallest permutation representation
►On 32 points
Generators in S32
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
001000
000100
0010160
00160016
,
1600000
0160000
0016000
001100
0016010
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
1370000
1040000
00160015
000011
00613016
007001
,
1040000
1370000
0010150
000011
0010160
00161610

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[13,10,0,0,0,0,7,4,0,0,0,0,0,0,16,0,6,7,0,0,0,0,13,0,0,0,0,1,0,0,0,0,15,1,16,1],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,1,16,0,0,0,0,0,16,0,0,15,1,16,1,0,0,0,1,0,0] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4J4K···4V8A···8H
order12222···2222244444···44···48···8
size11112···2444411112···24···44···4

44 irreducible representations

dim111111111224
type++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4D8⋊C22
kernelC24.98D4C23.24D4C23.36D4C23.37D4C23.38D4C2×C42⋊C2C22×M4(2)C22×C4○D4C2×C4○D4C22×C4C24C2
# reps1442211116714

In GAP, Magma, Sage, TeX 

C_2^4._{98}D_4
 
% in TeX
 
G:=Group("C2^4.98D4");
 
// GroupNames label
 
G:=SmallGroup(128,1628);
 
// by ID
 
G=gap.SmallGroup(128,1628);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,248,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,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*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
 
// generators/relations
 

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