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G = C24.26D4  order 128 = 27

26th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.26D4, C24.172C23, (C2×D4).76D4, C22.39(C4×D4), (C22×C4).25D4, C23.9D4⋊7C2, C23.565(C2×D4), C22.D4⋊4C4, C23.34D4⋊1C2, C23.68(C22×C4), (C23×C4).21C22, C22.104C22≀C2, C23.119(C4○D4), C22.46(C4⋊D4), C23.11(C22⋊C4), C22.11C24.6C2, C2.3(C23.7D4), (C22×D4).23C22, C2.29(C23.23D4), C22.53(C22.D4), C22⋊C4⋊6(C2×C4), (C22×C4)⋊10(C2×C4), (C2×D4).79(C2×C4), (C2×C23⋊C4).6C2, (C2×C4).15(C22⋊C4), C22.45(C2×C22⋊C4), (C2×C22⋊C4).11C22, (C2×C22.D4).2C2, SmallGroup(128,622)

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

Derived series
C1 — C23 — C24.26D4
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C22.11C24 — C24.26D4
Lower central
C1 — C2 — C23 — C24.26D4
Upper central
C1 — C22 — C24 — C24.26D4
Jennings
C1 — C2 — C24 — C24.26D4

Generators and relations for C24.26D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, eae-1=faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=be-1 >

Subgroups: 436 in 193 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C22.D4, C22.D4, C23×C4, C22×D4, C23.9D4, C23.34D4, C2×C23⋊C4, C22.11C24, C2×C22.D4, C24.26D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C23.7D4, C24.26D4

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

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A···4N4O···4T
order12222···2224···44···4
size11112···2444···48···8

32 irreducible representations

dim111111122224
type+++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4C23.7D4
kernelC24.26D4C23.9D4C23.34D4C2×C23⋊C4C22.11C24C2×C22.D4C22.D4C22×C4C2×D4C24C23C2
# reps122111852144

Matrix representation of C24.26D4 ►in GL6(𝔽5)

400000
040000
001000
000400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
030000
200000
000020
000002
000300
003000
,
010000
100000
000300
003000
000030
000003

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,2,0,0,0,0,0,0,2,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;
 

C24.26D4 in GAP, Magma, Sage, TeX 

C_2^4._{26}D_4
 
% in TeX
 
G:=Group("C2^4.26D4");
 
// GroupNames label
 
G:=SmallGroup(128,622);
 
// by ID
 
G=gap.SmallGroup(128,622);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,521,2804,1411,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^-1>;
 
// generators/relations
 

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