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G = C24.144D4order 128 = 27

13rd non-split extension by C24 of D4 acting via D4/C4=C2

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

Aliases: C24.144D4, (C2×C8)⋊38D4, C4(C87D4), C4(C88D4), C87D441C2, C88D455C2, (C23×C8)⋊10C2, C8.118(C2×D4), C4(C8.18D4), (C2×D8)⋊43C22, C221(C4○D8), C4⋊C4.19C23, C2.D846C22, C4.Q855C22, C8.18D441C2, (C2×C8).591C23, (C2×C4).254C24, (C2×Q16)⋊43C22, (C2×D4).58C23, C4.148(C22×D4), C23.383(C2×D4), (C22×C4).564D4, (C2×Q8).46C23, C4.211(C4⋊D4), D4⋊C458C22, C22.19C246C2, Q8⋊C459C22, (C2×SD16)⋊76C22, C23.24D44C2, C23.25D43C2, C4⋊D4.147C22, C22.34(C4⋊D4), (C23×C4).703C22, (C22×C8).557C22, C22.514(C22×D4), C22⋊Q8.152C22, (C22×C4).1533C23, C42⋊C2.106C22, (C2×C4○D8)⋊7C2, (C2×C4)(C88D4), (C2×C4)(C87D4), C2.16(C2×C4○D8), C4.21(C2×C4○D4), C2.72(C2×C4⋊D4), (C2×C4)(C8.18D4), (C2×C4).1426(C2×D4), (C2×C4).700(C4○D4), (C2×C4○D4).123C22, SmallGroup(128,1782)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.144D4
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C24.144D4
C1C2C2×C4 — C24.144D4
C1C2×C4C23×C4 — C24.144D4
C1C2C2C2×C4 — C24.144D4

Generators and relations for C24.144D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 484 in 260 conjugacy classes, 104 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×10], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C23×C4, C2×C4○D4 [×2], C23.24D4 [×2], C23.25D4, C88D4 [×4], C87D4 [×2], C8.18D4 [×2], C22.19C24 [×2], C23×C8, C2×C4○D8, C24.144D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C4○D8 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C4○D8 [×2], C24.144D4

Smallest permutation representation of C24.144D4
On 32 points
Generators in S32
(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(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 25 5 29)(2 28 6 32)(3 31 7 27)(4 26 8 30)(9 24 13 20)(10 19 14 23)(11 22 15 18)(12 17 16 21)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4J4K···4P8A···8P
order12222···22244444···44···48···8
size11112···28811112···28···82···2

44 irreducible representations

dim11111111122222
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D8
kernelC24.144D4C23.24D4C23.25D4C88D4C87D4C8.18D4C22.19C24C23×C8C2×C4○D8C2×C8C22×C4C24C2×C4C22
# reps121422211431416

Matrix representation of C24.144D4 in GL4(𝔽17) generated by

1000
0100
0010
00016
,
16000
0100
0010
00016
,
1000
0100
00160
00016
,
16000
01600
00160
00016
,
2000
0800
00150
0009
,
0800
2000
0009
00150
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[2,0,0,0,0,8,0,0,0,0,15,0,0,0,0,9],[0,2,0,0,8,0,0,0,0,0,0,15,0,0,9,0] >;

C24.144D4 in GAP, Magma, Sage, TeX

C_2^4._{144}D_4
% in TeX

G:=Group("C2^4.144D4");
// GroupNames label

G:=SmallGroup(128,1782);
// by ID

G=gap.SmallGroup(128,1782);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,248,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*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=e^3>;
// generators/relations

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