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G = C24.72D4order 128 = 27

27th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.72D4, M4(2)⋊18D4, C222C4≀C2, C4⋊D49C4, C4.58(C4×D4), C22⋊Q89C4, C426C420C2, C4.123C22≀C2, C23.557(C2×D4), (C22×C4).285D4, C22.41(C4⋊D4), (C22×M4(2))⋊10C2, C22.19C24.5C2, (C2×C42).275C22, (C23×C4).253C22, C23.118(C22⋊C4), C42⋊C2.15C22, (C22×C4).1358C23, C2.41(C42⋊C22), C2.10(C23.23D4), C4.136(C22.D4), (C2×M4(2)).317C22, (C2×C4≀C2)⋊12C2, C2.41(C2×C4≀C2), C4⋊C4.68(C2×C4), (C4×C22⋊C4)⋊23C2, (C2×D4).70(C2×C4), (C2×Q8).61(C2×C4), (C2×C4).1530(C2×D4), (C2×C4).754(C4○D4), (C2×C4).376(C22×C4), (C22×C4).275(C2×C4), (C2×C4○D4).15C22, (C2×C4).190(C22⋊C4), C22.262(C2×C22⋊C4), SmallGroup(128,603)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.72D4
C1C2C22C23C22×C4C23×C4C22×M4(2) — C24.72D4
C1C2C2×C4 — C24.72D4
C1C2×C4C23×C4 — C24.72D4
C1C2C2C22×C4 — C24.72D4

Generators and relations for C24.72D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, faf=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=cde3 >

Subgroups: 380 in 188 conjugacy classes, 58 normal (36 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×14], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×23], D4 [×8], Q8 [×2], C23 [×3], C23 [×6], C42 [×5], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×6], C22×C4 [×7], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C24, C2.C42, C4≀C2 [×4], C2×C42 [×2], C2×C22⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8, C2×M4(2) [×2], C2×M4(2) [×5], C23×C4, C2×C4○D4, C426C4 [×2], C4×C22⋊C4, C2×C4≀C2 [×2], C22.19C24, C22×M4(2), C24.72D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C2×C4≀C2, C42⋊C22, C24.72D4

Smallest permutation representation of C24.72D4
On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)
(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 12)(2 21)(3 14)(4 23)(5 16)(6 17)(7 10)(8 19)(9 32)(11 26)(13 28)(15 30)(18 29)(20 31)(22 25)(24 27)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I···4Q4R4S4T8A···8H
order1222222222444444444···44448···8
size1111222248111122224···48884···4

38 irreducible representations

dim11111111222224
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4D4C4○D4C4≀C2C42⋊C22
kernelC24.72D4C426C4C4×C22⋊C4C2×C4≀C2C22.19C24C22×M4(2)C4⋊D4C22⋊Q8M4(2)C22×C4C24C2×C4C22C2
# reps12121144431482

Matrix representation of C24.72D4 in GL4(𝔽17) generated by

0100
1000
0010
0001
,
16000
01600
0010
0001
,
16000
01600
0010
00016
,
1000
0100
00160
00016
,
13000
01300
0001
0040
,
0400
13000
00016
00160
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,4,0,0,1,0],[0,13,0,0,4,0,0,0,0,0,0,16,0,0,16,0] >;

C24.72D4 in GAP, Magma, Sage, TeX

C_2^4._{72}D_4
% in TeX

G:=Group("C2^4.72D4");
// GroupNames label

G:=SmallGroup(128,603);
// by ID

G=gap.SmallGroup(128,603);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2028]);
// Polycyclic

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

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