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G = C22.94C25order 128 = 27

75th central stem extension by C22 of C25

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

Aliases: C22.94C25, C42.86C23, C24.137C23, C23.138C24, C4.802+ 1+4, D45D420C2, Q85D418C2, (C4×D4)⋊46C22, C234(C4○D4), C233D48C2, (C2×C4).84C24, (C4×Q8)⋊45C22, C4⋊C4.490C23, C4⋊D482C22, (C23×C4)⋊43C22, (C2×C42)⋊61C22, C22⋊Q834C22, C22≀C234C22, C22.32C244C2, C422C24C22, (C2×D4).303C23, C4.4D484C22, (C22×D4)⋊38C22, (C2×Q8).450C23, C42.C255C22, (C22×Q8)⋊34C22, C22.45C246C2, C22.19C2430C2, C42⋊C240C22, C22.11C2418C2, C22⋊C4.104C23, (C22×C4).365C23, C22.D48C22, C2.35(C2×2+ 1+4), C2.28(C2.C25), C22.33C244C2, C23.36C2329C2, C22.47C2415C2, C22.46C2416C2, (C2×C4×D4)⋊91C2, (C2×C4⋊D4)⋊68C2, (C2×C4⋊C4)⋊75C22, (C2×C22⋊Q8)⋊78C2, C22⋊C4(C22⋊Q8), (C2×C4○D4)⋊32C22, C2.50(C22×C4○D4), C22.35(C2×C4○D4), (C2×C22⋊C4)⋊50C22, SmallGroup(128,2237)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.94C25
C1C2C22C23C24C23×C4C2×C4×D4 — C22.94C25
C1C22 — C22.94C25
C1C22 — C22.94C25
C1C22 — C22.94C25

Generators and relations for C22.94C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=b, ab=ba, dcd-1=gcg=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 576 conjugacy classes, 390 normal (38 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×21], C22, C22 [×6], C22 [×36], C2×C4 [×4], C2×C4 [×18], C2×C4 [×35], D4 [×38], Q8 [×6], C23 [×3], C23 [×10], C23 [×14], C42 [×12], C22⋊C4 [×48], C4⋊C4 [×36], C22×C4 [×9], C22×C4 [×18], C22×C4 [×4], C2×D4 [×24], C2×D4 [×14], C2×Q8 [×4], C2×Q8 [×2], C4○D4 [×4], C24 [×2], C24 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C42⋊C2 [×10], C4×D4 [×30], C4×Q8 [×2], C22≀C2 [×8], C4⋊D4 [×22], C22⋊Q8 [×14], C22.D4 [×20], C4.4D4 [×6], C42.C2 [×6], C422C2 [×12], C23×C4 [×2], C22×D4 [×2], C22×D4 [×2], C22×Q8, C2×C4○D4 [×2], C2×C4×D4, C22.11C24 [×2], C2×C4⋊D4, C2×C22⋊Q8, C22.19C24 [×2], C23.36C23 [×2], C233D4 [×2], C22.32C24 [×2], C22.33C24 [×2], D45D4 [×2], Q85D4 [×2], C22.45C24 [×4], C22.46C24 [×2], C22.47C24 [×6], C22.94C25
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, C22.94C25

Smallest permutation representation of C22.94C25
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 23)(2 32)(3 21)(4 30)(5 10)(6 15)(7 12)(8 13)(9 19)(11 17)(14 20)(16 18)(22 26)(24 28)(25 29)(27 31)
(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 21 3 23)(2 22 4 24)(5 10 7 12)(6 11 8 9)(13 17 15 19)(14 18 16 20)(25 31 27 29)(26 32 28 30)
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2O4A···4L4M···4AB
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111111111244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+4C2.C25
kernelC22.94C25C2×C4×D4C22.11C24C2×C4⋊D4C2×C22⋊Q8C22.19C24C23.36C23C233D4C22.32C24C22.33C24D45D4Q85D4C22.45C24C22.46C24C22.47C24C23C4C2
# reps112112222222426822

Matrix representation of C22.94C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
120000
040000
002210
001344
000002
000030
,
200000
020000
002000
000001
002133
000400
,
120000
440000
003000
004211
000003
000030
,
400000
040000
001033
000100
000040
000004
,
100000
010000
001000
003422
000001
000010

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

C22.94C25 in GAP, Magma, Sage, TeX

C_2^2._{94}C_2^5
% in TeX

G:=Group("C2^2.94C2^5");
// GroupNames label

G:=SmallGroup(128,2237);
// by ID

G=gap.SmallGroup(128,2237);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,1684]);
// Polycyclic

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