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G = C24.183D4order 128 = 27

38th non-split extension by C24 of D4 acting via D4/C22=C2

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

Aliases: C24.183D4, C4.Q87C22, C4⋊C4.52C23, (C2×C8).32C23, C2.D818C22, (C2×C4).290C24, C24.4C47C2, (C2×D4).79C23, C23.665(C2×D4), (C22×C4).441D4, (C2×Q8).67C23, D4⋊C418C22, Q8⋊C420C22, C22.D811C2, C23.36D49C2, C23.46D41C2, C23.47D41C2, C22⋊C8.14C22, M4(2)⋊C422C2, C4⋊D4.155C22, C23.48D411C2, C22.29(C8⋊C22), (C23×C4).560C22, C22.550(C22×D4), C22⋊Q8.160C22, (C22×C4).1006C23, C22.19C24.18C2, C22.18(C8.C22), (C2×M4(2)).72C22, C42⋊C2.123C22, C4.126(C22.D4), C22.40(C22.D4), (C22×C4⋊C4)⋊35C2, C4.100(C2×C4○D4), (C2×C4).486(C2×D4), C2.28(C2×C8⋊C22), C2.28(C2×C8.C22), (C2×C4).485(C4○D4), (C2×C4⋊C4).927C22, (C2×C4○D4).137C22, C2.55(C2×C22.D4), SmallGroup(128,1824)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.183D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — C24.183D4
C1C2C2×C4 — C24.183D4
C1C22C23×C4 — C24.183D4
C1C2C2C2×C4 — C24.183D4

Generators and relations for C24.183D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 436 in 229 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×9], C22, C22 [×6], C22 [×17], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×31], D4 [×8], Q8 [×2], C23 [×3], C23 [×7], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×15], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C4⋊C4 [×6], C2×C4⋊C4 [×3], C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C23×C4, C23×C4, C2×C4○D4, C24.4C4, C23.36D4 [×2], M4(2)⋊C4 [×2], C22.D8 [×2], C23.46D4 [×2], C23.47D4 [×2], C23.48D4 [×2], C22×C4⋊C4, C22.19C24, C24.183D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C2×C22.D4, C2×C8⋊C22, C2×C8.C22, C24.183D4

Smallest permutation representation of C24.183D4
On 32 points
Generators in S32
(2 27)(4 29)(6 31)(8 25)(9 22)(11 24)(13 18)(15 20)
(1 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 22)(2 16)(3 20)(4 14)(5 18)(6 12)(7 24)(8 10)(9 26)(11 32)(13 30)(15 28)(17 31)(19 29)(21 27)(23 25)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J4A4B4C4D4E···4N4O4P4Q8A8B8C8D
order12222···2244444···44448888
size11112···2822224···48888888

32 irreducible representations

dim111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22C8.C22
kernelC24.183D4C24.4C4C23.36D4M4(2)⋊C4C22.D8C23.46D4C23.47D4C23.48D4C22×C4⋊C4C22.19C24C22×C4C24C2×C4C22C22
# reps112222221131822

Matrix representation of C24.183D4 in GL6(𝔽17)

100000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0015910
0015901
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
040000
1300000
0041658
001214715
00615160
0028160
,
010000
100000
0028150
0000161
00108114
00109114

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,15,15,0,0,0,16,9,9,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,4,12,6,2,0,0,16,14,15,8,0,0,5,7,16,16,0,0,8,15,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,10,10,0,0,8,0,8,9,0,0,15,16,11,11,0,0,0,1,4,4] >;

C24.183D4 in GAP, Magma, Sage, TeX

C_2^4._{183}D_4
% in TeX

G:=Group("C2^4.183D4");
// GroupNames label

G:=SmallGroup(128,1824);
// by ID

G=gap.SmallGroup(128,1824);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,2019,4037,1027,124]);
// 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=f*a*f=a*c=c*a,a*d=d*a,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=c*d*e^3>;
// generators/relations

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