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G = C24.178D4order 128 = 27

33rd non-split extension by C24 of D4 acting via D4/C22=C2

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

Aliases: C24.178D4, (C2×Q8)⋊36D4, Q8⋊D41C2, C4⋊C4.6C23, (C2×C8).7C23, Q8.40(C2×D4), (Q8×C23)⋊6C2, C4.82C22≀C2, C4.41(C22×D4), (C2×C4).223C24, C24.4C43C2, C22⋊Q1611C2, (C2×Q16)⋊12C22, (C2×SD16)⋊2C22, (C2×D4).26C23, (C22×C4).420D4, C23.647(C2×D4), (C2×Q8).19C23, Q8⋊C411C22, C22.58C22≀C2, C23.38D42C2, C22⋊C8.11C22, C224(C8.C22), C4⋊D4.145C22, (C22×C4).961C23, (C23×C4).543C22, C22.483(C22×D4), C22⋊Q8.150C22, C22.19C24.15C2, C42⋊C2.95C22, (C2×M4(2)).39C22, (C22×Q8).467C22, (C2×C4).451(C2×D4), (C2×C8.C22)⋊5C2, C2.41(C2×C22≀C2), C2.10(C2×C8.C22), (C2×C4○D4).99C22, SmallGroup(128,1736)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.178D4
C1C2C22C2×C4C22×C4C23×C4Q8×C23 — C24.178D4
C1C2C2×C4 — C24.178D4
C1C22C23×C4 — C24.178D4
C1C2C2C2×C4 — C24.178D4

Generators and relations for C24.178D4
 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=e3 >

Subgroups: 660 in 369 conjugacy classes, 112 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×4], C4 [×13], C22, C22 [×6], C22 [×17], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×43], D4 [×8], Q8 [×8], Q8 [×30], C23, C23 [×2], C23 [×7], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×19], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×12], C2×Q8 [×50], C4○D4 [×4], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C2×SD16 [×4], C2×Q16 [×4], C8.C22 [×8], C23×C4, C23×C4, C22×Q8 [×6], C22×Q8 [×11], C2×C4○D4, C24.4C4, C23.38D4 [×2], Q8⋊D4 [×4], C22⋊Q16 [×4], C22.19C24, C2×C8.C22 [×2], Q8×C23, C24.178D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C8.C22 [×4], C22×D4 [×3], C2×C22≀C2, C2×C8.C22 [×2], C24.178D4

Smallest permutation representation of C24.178D4
On 32 points
Generators in S32
(2 30)(4 32)(6 26)(8 28)(10 18)(12 20)(14 22)(16 24)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 10)(2 13)(3 16)(4 11)(5 14)(6 9)(7 12)(8 15)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111112224
type+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4C8.C22
kernelC24.178D4C24.4C4C23.38D4Q8⋊D4C22⋊Q16C22.19C24C2×C8.C22Q8×C23C22×C4C2×Q8C24C22
# reps112441213814

Matrix representation of C24.178D4 in GL6(𝔽17)

100000
13160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
000010
0091301
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
15160000
520000
0000130
00151649
000400
0011091
,
15160000
320000
000010
0084115
001000
00816613

G:=sub<GL(6,GF(17))| [1,13,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,9,0,0,0,16,0,13,0,0,0,0,1,0,0,0,0,0,0,1],[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],[15,5,0,0,0,0,16,2,0,0,0,0,0,0,0,15,0,11,0,0,0,16,4,0,0,0,13,4,0,9,0,0,0,9,0,1],[15,3,0,0,0,0,16,2,0,0,0,0,0,0,0,8,1,8,0,0,0,4,0,16,0,0,1,1,0,6,0,0,0,15,0,13] >;

C24.178D4 in GAP, Magma, Sage, TeX

C_2^4._{178}D_4
% in TeX

G:=Group("C2^4.178D4");
// GroupNames label

G:=SmallGroup(128,1736);
// by ID

G=gap.SmallGroup(128,1736);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,2804,1411,172]);
// 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=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=e^3>;
// generators/relations

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