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G = C25.3C4order 128 = 27

3rd non-split extension by C25 of C4 acting faithfully

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

Aliases: C25.3C4, C234M4(2), C23⋊C812C2, (C23×C4).8C4, C24.21(C2×C4), C22⋊C840C22, (C22×C4).198D4, C24.4C415C2, C22.25(C23⋊C4), C2.6(C24.4C4), (C22×C4).426C23, (C23×C4).197C22, C23.163(C22×C4), C22.15(C2×M4(2)), C23.165(C22⋊C4), C22.16(C4.D4), C2.6(C2×C23⋊C4), C2.5(C2×C4.D4), (C2×C4).1123(C2×D4), (C2×C22⋊C4).34C4, (C2×C4).68(C22⋊C4), (C22×C4).175(C2×C4), (C22×C22⋊C4).14C2, C22.144(C2×C22⋊C4), (C2×C22⋊C4).402C22, SmallGroup(128,194)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C25.3C4
C1C2C22C2×C4C22×C4C23×C4C22×C22⋊C4 — C25.3C4
C1C2C23 — C25.3C4
C1C22C23×C4 — C25.3C4
C1C2C22C22×C4 — C25.3C4

Generators and relations for C25.3C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, faf-1=acd, bc=cb, bd=db, fbf-1=be=eb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 564 in 220 conjugacy classes, 52 normal (16 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C22, C22 [×6], C22 [×46], C8 [×4], C2×C4 [×4], C2×C4 [×16], C23 [×3], C23 [×4], C23 [×46], C22⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×6], C22×C4 [×6], C24, C24 [×2], C24 [×10], C22⋊C8 [×4], C22⋊C8 [×2], C2×C22⋊C4 [×4], C2×C22⋊C4 [×4], C2×M4(2) [×2], C23×C4 [×2], C25, C23⋊C8 [×4], C24.4C4 [×2], C22×C22⋊C4, C25.3C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C24.4C4, C2×C23⋊C4, C2×C4.D4, C25.3C4

Permutation representations of C25.3C4
On 16 points - transitive group 16T252
Generators in S16
(1 5)(2 14)(3 11)(6 10)(7 15)(9 13)
(1 9)(2 14)(3 11)(4 16)(5 13)(6 10)(7 15)(8 12)
(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

G:=sub<Sym(16)| (1,5)(2,14)(3,11)(6,10)(7,15)(9,13), (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;

G:=Group( (1,5)(2,14)(3,11)(6,10)(7,15)(9,13), (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );

G=PermutationGroup([(1,5),(2,14),(3,11),(6,10),(7,15),(9,13)], [(1,9),(2,14),(3,11),(4,16),(5,13),(6,10),(7,15),(8,12)], [(1,5),(2,10),(3,7),(4,12),(6,14),(8,16),(9,13),(11,15)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,252);

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4J8A···8H
order12222···2222244444···48···8
size11112···2444422224···48···8

32 irreducible representations

dim11111112244
type+++++++
imageC1C2C2C2C4C4C4D4M4(2)C23⋊C4C4.D4
kernelC25.3C4C23⋊C8C24.4C4C22×C22⋊C4C2×C22⋊C4C23×C4C25C22×C4C23C22C22
# reps14214224822

Matrix representation of C25.3C4 in GL6(𝔽17)

100000
010000
001000
0001600
0000160
000001
,
100000
0160000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
010000
400000
000010
000001
000100
0016000

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C25.3C4 in GAP, Magma, Sage, TeX

C_2^5._3C_4
% in TeX

G:=Group("C2^5.3C4");
// GroupNames label

G:=SmallGroup(128,194);
// by ID

G=gap.SmallGroup(128,194);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,851,172]);
// Polycyclic

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

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