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G = C22.78C25order 128 = 27

59th central stem extension by C22 of C25

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

Aliases: C22.78C25, C42.80C23, C23.130C24, C24.503C23, C2222- 1+4, C4○D418D4, (D4×Q8)⋊16C2, Q8(C4⋊D4), D4(C22⋊Q8), D4.58(C2×D4), C4⋊Q830C22, Q8.59(C2×D4), D45D415C2, D46D418C2, Q85D415C2, (C4×D4)⋊36C22, (C2×C4).71C24, (C4×Q8)⋊38C22, C2.30(D4×C23), C4⋊C4.292C23, C4⋊D421C22, C4.119(C22×D4), C22⋊Q895C22, (C2×D4).299C23, C4.4D424C22, C22⋊C4.18C23, (C2×2- 1+4)⋊7C2, (C2×Q8).443C23, (C22×Q8)⋊31C22, C22.12(C22×D4), C22.19C2425C2, C42⋊C233C22, C22≀C2.26C22, (C23×C4).605C22, (C22×C4).352C23, C22.D43C22, C2.19(C2×2- 1+4), C2.17(C2.C25), (C22×D4).598C22, C23.38C2320C2, C23.33C2316C2, C22.31C2412C2, (C2×C4⋊C4)⋊69C22, (C2×C4).666(C2×D4), (C2×C22⋊Q8)⋊75C2, (C22×C4○D4)⋊24C2, (C2×C4○D4)⋊26C22, (C2×C22⋊C4).379C22, SmallGroup(128,2221)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.78C25
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C22.78C25
C1C22 — C22.78C25
C1C22 — C22.78C25
C1C22 — C22.78C25

Generators and relations for C22.78C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=e2=f2=1, c2=g2=a, ab=ba, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1156 in 761 conjugacy classes, 430 normal (20 characteristic)
C1, C2 [×3], C2 [×13], C4 [×8], C4 [×18], C22, C22 [×8], C22 [×35], C2×C4 [×2], C2×C4 [×32], C2×C4 [×56], D4 [×12], D4 [×42], Q8 [×4], Q8 [×24], C23, C23 [×8], C23 [×15], C42 [×6], C22⋊C4 [×38], C4⋊C4, C4⋊C4 [×33], C22×C4, C22×C4 [×33], C22×C4 [×12], C2×D4, C2×D4 [×26], C2×D4 [×12], C2×Q8, C2×Q8 [×12], C2×Q8 [×24], C4○D4 [×8], C4○D4 [×64], C24 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×9], C42⋊C2 [×3], C4×D4 [×18], C4×Q8 [×2], C22≀C2 [×6], C4⋊D4, C4⋊D4 [×21], C22⋊Q8 [×30], C22.D4 [×18], C4.4D4 [×6], C4⋊Q8 [×6], C23×C4 [×3], C22×D4 [×3], C22×Q8, C22×Q8 [×5], C2×C4○D4 [×2], C2×C4○D4 [×15], C2×C4○D4 [×8], 2- 1+4 [×8], C23.33C23, C2×C22⋊Q8 [×3], C22.19C24 [×3], C23.38C23 [×3], C22.31C24 [×3], D45D4 [×6], D46D4 [×6], Q85D4 [×2], D4×Q8 [×2], C22×C4○D4, C2×2- 1+4, C22.78C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2- 1+4 [×2], C25, D4×C23, C2×2- 1+4, C2.C25, C22.78C25

Smallest permutation representation of C22.78C25
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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(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 7)(2 6)(3 5)(4 8)(9 23)(10 22)(11 21)(12 24)(13 31)(14 30)(15 29)(16 32)(17 28)(18 27)(19 26)(20 25)
(1 3)(2 26)(4 28)(5 7)(6 19)(8 17)(9 11)(10 32)(12 30)(13 15)(14 24)(16 22)(18 20)(21 23)(25 27)(29 31)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 29)(26 30)(27 31)(28 32)
(1 21 3 23)(2 24 4 22)(5 9 7 11)(6 12 8 10)(13 25 15 27)(14 28 16 26)(17 32 19 30)(18 31 20 29)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2K2L···2P4A···4J4K···4AA
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111111244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D42- 1+4C2.C25
kernelC22.78C25C23.33C23C2×C22⋊Q8C22.19C24C23.38C23C22.31C24D45D4D46D4Q85D4D4×Q8C22×C4○D4C2×2- 1+4C4○D4C22C2
# reps113333662211822

Matrix representation of C22.78C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
120000
040000
000010
000001
004000
000400
,
400000
040000
000300
002000
000002
000030
,
100000
440000
004000
000400
000010
000001
,
100000
010000
000100
001000
000001
000010
,
100000
010000
003000
000300
000020
000002

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[1,4,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;

C22.78C25 in GAP, Magma, Sage, TeX

C_2^2._{78}C_2^5
% in TeX

G:=Group("C2^2.78C2^5");
// GroupNames label

G:=SmallGroup(128,2221);
// by ID

G=gap.SmallGroup(128,2221);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,570,1684,102]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=e^2=f^2=1,c^2=g^2=a,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=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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