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G = C22.103C25order 128 = 27

84th central stem extension by C22 of C25

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.94C23, C24.140C23, C23.141C24, C22.103C25, C4.1582+ 1+4, D4216C2, (D4×Q8)⋊21C2, C4⋊Q894C22, D414(C4○D4), D45D423C2, Q85D420C2, D42(C4.4D4), (C4×D4)⋊51C22, (C2×C4).93C24, (C4×Q8)⋊50C22, C4⋊D429C22, C4⋊C4.497C23, (C2×C42)⋊65C22, C22⋊Q837C22, C22≀C211C22, C22.32C247C2, (C2×D4).476C23, C4.4D430C22, C22⋊C4.27C23, (C2×Q8).455C23, C42.C276C22, (C22×Q8)⋊35C22, C42⋊C244C22, C422C239C22, C22.11C2422C2, C22.45C249C2, C41D4.188C22, (C22×C4).373C23, C2.40(C2×2+ 1+4), C2.34(C2.C25), C22.26C2443C2, (C22×D4).430C22, C22.D454C22, C22.49C2415C2, C22.36C2417C2, C22.50C2425C2, C23.36C2334C2, C22.53C2416C2, (C4×C4○D4)⋊33C2, C4⋊C4(C42.C2), C4.276(C2×C4○D4), (C2×D4)(C4.4D4), (C2×C4.4D4)⋊56C2, (C2×C4○D4)⋊36C22, C22⋊C4(C4.4D4), C22.42(C2×C4○D4), C2.59(C22×C4○D4), (C2×C22⋊C4)⋊52C22, SmallGroup(128,2246)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.103C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.103C25
C1C22 — C22.103C25
C1C22 — C22.103C25
C1C22 — C22.103C25

Generators and relations for C22.103C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, g2=b, ab=ba, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 916 in 571 conjugacy classes, 390 normal (50 characteristic)
C1, C2 [×3], C2 [×11], C4 [×4], C4 [×20], C22, C22 [×4], C22 [×33], C2×C4 [×6], C2×C4 [×16], C2×C4 [×31], D4 [×4], D4 [×34], Q8 [×12], C23, C23 [×8], C23 [×16], C42 [×4], C42 [×12], C22⋊C4 [×52], C4⋊C4 [×4], C4⋊C4 [×24], C22×C4 [×3], C22×C4 [×20], C2×D4 [×2], C2×D4 [×22], C2×D4 [×12], C2×Q8 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×8], C24 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×12], C42⋊C2, C42⋊C2 [×10], C4×D4 [×3], C4×D4 [×24], C4×Q8 [×3], C4×Q8 [×2], C22≀C2 [×8], C4⋊D4, C4⋊D4 [×16], C22⋊Q8, C22⋊Q8 [×12], C22.D4 [×14], C4.4D4, C4.4D4 [×20], C42.C2, C422C2 [×10], C41D4, C4⋊Q8, C4⋊Q8 [×2], C22×D4 [×4], C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×2], C4×C4○D4, C22.11C24 [×2], C2×C4.4D4 [×2], C23.36C23, C23.36C23 [×2], C22.26C24, C22.32C24 [×4], C22.36C24 [×2], D42, D45D4 [×4], Q85D4 [×2], D4×Q8, C22.45C24 [×4], C22.49C24 [×2], C22.50C24, C22.53C24, C22.103C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×2], C25, C22×C4○D4, C2×2+ 1+4, C2.C25, C22.103C25

Smallest permutation representation of C22.103C25
On 32 points
Generators in S32
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 15)(2 16)(3 13)(4 14)(5 19)(6 20)(7 17)(8 18)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 32)
(1 8)(2 19)(3 6)(4 17)(5 16)(7 14)(9 29)(10 22)(11 31)(12 24)(13 20)(15 18)(21 25)(23 27)(26 30)(28 32)
(1 23)(2 24)(3 21)(4 22)(5 26)(6 27)(7 28)(8 25)(9 18)(10 19)(11 20)(12 17)(13 29)(14 30)(15 31)(16 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)
(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 19 15 5)(2 20 16 6)(3 17 13 7)(4 18 14 8)(9 30 25 22)(10 31 26 23)(11 32 27 24)(12 29 28 21)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2N4A···4N4O···4AC
order122222222···24···44···4
size111122224···42···24···4

44 irreducible representations

dim1111111111111111244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+4C2.C25
kernelC22.103C25C4×C4○D4C22.11C24C2×C4.4D4C23.36C23C22.26C24C22.32C24C22.36C24D42D45D4Q85D4D4×Q8C22.45C24C22.49C24C22.50C24C22.53C24D4C4C2
# reps1122314214214211822

Matrix representation of C22.103C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
410000
010000
000220
003021
000034
000032
,
400000
040000
000110
000100
001400
000404
,
230000
430000
002000
000200
000020
000002
,
400000
040000
001002
000102
000040
000004
,
300000
030000
000110
001010
000040
000041

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,2,2,3,3,0,0,0,1,4,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,1,1,4,4,0,0,1,0,0,0,0,0,0,0,0,4],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,2,2,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,1,4,4,0,0,0,0,0,1] >;

C22.103C25 in GAP, Magma, Sage, TeX

C_2^2._{103}C_2^5
% in TeX

G:=Group("C2^2.103C2^5");
// GroupNames label

G:=SmallGroup(128,2246);
// by ID

G=gap.SmallGroup(128,2246);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,184,570,1684,172]);
// Polycyclic

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

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