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G = C22.70C25order 128 = 27

51st central stem extension by C22 of C25

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

Aliases: C22.70C25, C23.129C24, C24.500C23, C42.567C23, C4.1862+ 1+4, C42D42, D4227C2, D43(C4×D4), C4⋊Q888C22, C43(D45D4), D411(C4○D4), D45D444C2, C43(D43Q8), D43Q849C2, (C4×Q8)⋊98C22, (C4×D4)⋊110C22, C4⋊C4.479C23, C4⋊D477C22, (C2×C4).601C24, (C23×C4)⋊37C22, (C2×C42)⋊55C22, C22⋊Q892C22, C22≀C233C22, (C2×D4).297C23, C4.4D477C22, C22⋊C4.92C23, (C2×Q8).282C23, C42.C251C22, C22.19C2419C2, C422C233C22, C42⋊C229C22, C41D4.183C22, (C22×C4).619C23, C42(C22.45C24), C22.45C2428C2, C2.24(C2×2+ 1+4), C22.26C2431C2, (C22×D4).596C22, C22.D447C22, C42(C22.53C24), C43(C22.47C24), C22.53C2429C2, C22.47C2445C2, C23.36C2323C2, (C2×C4×D4)⋊89C2, (C4×C4○D4)⋊23C2, C4.272(C2×C4○D4), (C2×C4⋊C4)⋊140C22, (C2×C4○D4)⋊21C22, C2.41(C22×C4○D4), C22.39(C2×C4○D4), (C2×C22⋊C4)⋊89C22, (C2×C4)(C22.53C24), (C2×C4)(C22.47C24), SmallGroup(128,2213)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.70C25
C1C2C22C2×C4C22×C4C23×C4C2×C4×D4 — C22.70C25
C1C22 — C22.70C25
C1C2×C4 — C22.70C25
C1C22 — C22.70C25

Generators and relations for C22.70C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=ba=ab, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 956 in 621 conjugacy classes, 400 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×13], C4 [×6], C4 [×19], C22, C22 [×8], C22 [×39], C2×C4 [×2], C2×C4 [×20], C2×C4 [×50], D4 [×8], D4 [×38], Q8 [×6], C23, C23 [×8], C23 [×20], C42 [×2], C42 [×14], C22⋊C4 [×44], C4⋊C4 [×32], C22×C4, C22×C4 [×30], C22×C4 [×8], C2×D4 [×24], C2×D4 [×16], C2×Q8 [×4], C4○D4 [×12], C24 [×4], C2×C42, C2×C42 [×4], C2×C22⋊C4 [×8], C2×C4⋊C4 [×4], C42⋊C2 [×10], C4×D4 [×40], C4×Q8 [×4], C22≀C2 [×8], C4⋊D4 [×16], C22⋊Q8 [×12], C22.D4 [×16], C4.4D4 [×6], C42.C2 [×4], C422C2 [×8], C41D4, C4⋊Q8, C23×C4 [×4], C22×D4 [×4], C2×C4○D4 [×4], C2×C4×D4 [×4], C4×C4○D4 [×2], C22.19C24 [×4], C23.36C23 [×4], C22.26C24, D42, D45D4 [×4], C22.45C24 [×4], C22.47C24 [×4], D43Q8 [×2], C22.53C24, C22.70C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×8], C24 [×31], C2×C4○D4 [×12], 2+ 1+4 [×2], C25, C22×C4○D4 [×2], C2×2+ 1+4, C22.70C25

Smallest permutation representation of C22.70C25
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 30)(6 31)(7 32)(8 29)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)
(1 11)(2 12)(3 9)(4 10)(5 32)(6 29)(7 30)(8 31)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)
(1 27)(2 28)(3 25)(4 26)(5 17)(6 18)(7 19)(8 20)(9 24)(10 21)(11 22)(12 23)(13 29)(14 30)(15 31)(16 32)
(1 19)(2 20)(3 17)(4 18)(5 22)(6 23)(7 24)(8 21)(9 16)(10 13)(11 14)(12 15)(25 30)(26 31)(27 32)(28 29)
(5 32)(6 29)(7 30)(8 31)(21 26)(22 27)(23 28)(24 25)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 26)(22 27)(23 28)(24 25)(29 31)(30 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)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D···2K2L···2P4A4B4C4D4E···4T4U···4AG
order12222···22···244444···44···4
size11112···24···411112···24···4

50 irreducible representations

dim11111111111124
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC22.70C25C2×C4×D4C4×C4○D4C22.19C24C23.36C23C22.26C24D42D45D4C22.45C24C22.47C24D43Q8C22.53C24D4C4
# reps142441144421162

Matrix representation of C22.70C25 in GL4(𝔽5) generated by

4000
0400
0010
0001
,
1000
0100
0040
0004
,
1000
0400
0001
0010
,
0100
1000
0040
0004
,
1000
0100
0010
0004
,
1000
0400
0040
0004
,
2000
0200
0030
0003
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3] >;

C22.70C25 in GAP, Magma, Sage, TeX

C_2^2._{70}C_2^5
% in TeX

G:=Group("C2^2.70C2^5");
// GroupNames label

G:=SmallGroup(128,2213);
// by ID

G=gap.SmallGroup(128,2213);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,242]);
// Polycyclic

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