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G = C22.104C25  order 128 = 27

85th central stem extension by C22 of C25

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

Aliases: C42.95C23, C22.104C25, C23.142C24, C4.1092- 1+4, D4⋊6D4⋊26C2, Q8⋊3Q8⋊20C2, D4⋊3Q8⋊26C2, (C2×C4).94C24, D4○2(C42.C2), D4.41(C4○D4), C4⋊C4.301C23, C4⋊Q8.221C22, (C2×D4).508C23, (C4×D4).238C22, C22⋊C4.28C23, (C2×Q8).456C23, (C4×Q8).225C22, C4⋊D4.116C22, (C22×C4).374C23, (C2×C42).953C22, C22⋊Q8.230C22, C2.29(C2×2- 1+4), C2.35(C2.C25), C42⋊2C2.17C22, C22.33C24⋊7C2, C4.4D4.187C22, C42.C2.169C22, C23.33C23⋊26C2, C23.36C23⋊35C2, C42⋊C2.233C22, C22.35C24⋊12C2, C22.47C24⋊20C2, C22.46C24⋊21C2, C23.37C23⋊44C2, C22.D4.31C22, (C4×C4○D4)⋊34C2, C4⋊C4○(C4.4D4), C4.277(C2×C4○D4), (C2×D4)○(C42.C2), (C2×C42.C2)⋊47C2, C2.60(C22×C4○D4), C22.43(C2×C4○D4), C22⋊C4○(C42.C2), (C2×C4⋊C4).710C22, (C2×C4○D4).232C22, SmallGroup(128,2247)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C22.104C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.104C25
Lower central
C1 — C22 — C22.104C25
Upper central
C1 — C22 — C22.104C25
Jennings
C1 — C22 — C22.104C25

Generators and relations for C22.104C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=f2=1, c2=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: 660 in 507 conjugacy classes, 390 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C2×C4○D4, C2×C4○D4, C4×C4○D4, C23.33C23, C2×C42.C2, C23.36C23, C23.36C23, C23.37C23, C22.33C24, C22.35C24, D4⋊6D4, D4⋊6D4, C22.46C24, C22.47C24, D4⋊3Q8, Q8⋊3Q8, C22.104C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22×C4○D4, C2×2- 1+4, C2.C25, C22.104C25

Smallest permutation representation of C22.104C25
►On 64 points
Generators in S64
(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)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)

(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)

(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 19)(2 18)(3 17)(4 20)(5 37)(6 40)(7 39)(8 38)(9 15)(10 14)(11 13)(12 16)(21 36)(22 35)(23 34)(24 33)(25 31)(26 30)(27 29)(28 32)(41 47)(42 46)(43 45)(44 48)(49 61)(50 64)(51 63)(52 62)(53 59)(54 58)(55 57)(56 60)

(1 13 3 15)(2 58 4 60)(5 41 7 43)(6 26 8 28)(9 19 11 17)(10 64 12 62)(14 50 16 52)(18 54 20 56)(21 31 23 29)(22 48 24 46)(25 34 27 36)(30 38 32 40)(33 42 35 44)(37 47 39 45)(49 59 51 57)(53 63 55 61)

(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)

(1 23 51 39)(2 22 52 38)(3 21 49 37)(4 24 50 40)(5 17 36 61)(6 20 33 64)(7 19 34 63)(8 18 35 62)(9 25 53 41)(10 28 54 44)(11 27 55 43)(12 26 56 42)(13 29 57 45)(14 32 58 48)(15 31 59 47)(16 30 60 46)
 
G:=sub<Sym(64)| (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)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,19)(2,18)(3,17)(4,20)(5,37)(6,40)(7,39)(8,38)(9,15)(10,14)(11,13)(12,16)(21,36)(22,35)(23,34)(24,33)(25,31)(26,30)(27,29)(28,32)(41,47)(42,46)(43,45)(44,48)(49,61)(50,64)(51,63)(52,62)(53,59)(54,58)(55,57)(56,60), (1,13,3,15)(2,58,4,60)(5,41,7,43)(6,26,8,28)(9,19,11,17)(10,64,12,62)(14,50,16,52)(18,54,20,56)(21,31,23,29)(22,48,24,46)(25,34,27,36)(30,38,32,40)(33,42,35,44)(37,47,39,45)(49,59,51,57)(53,63,55,61), (1,11)(2,12)(3,9)(4,10)(5,45)(6,46)(7,47)(8,48)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,23,51,39)(2,22,52,38)(3,21,49,37)(4,24,50,40)(5,17,36,61)(6,20,33,64)(7,19,34,63)(8,18,35,62)(9,25,53,41)(10,28,54,44)(11,27,55,43)(12,26,56,42)(13,29,57,45)(14,32,58,48)(15,31,59,47)(16,30,60,46)>;
 
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)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,19)(2,18)(3,17)(4,20)(5,37)(6,40)(7,39)(8,38)(9,15)(10,14)(11,13)(12,16)(21,36)(22,35)(23,34)(24,33)(25,31)(26,30)(27,29)(28,32)(41,47)(42,46)(43,45)(44,48)(49,61)(50,64)(51,63)(52,62)(53,59)(54,58)(55,57)(56,60), (1,13,3,15)(2,58,4,60)(5,41,7,43)(6,26,8,28)(9,19,11,17)(10,64,12,62)(14,50,16,52)(18,54,20,56)(21,31,23,29)(22,48,24,46)(25,34,27,36)(30,38,32,40)(33,42,35,44)(37,47,39,45)(49,59,51,57)(53,63,55,61), (1,11)(2,12)(3,9)(4,10)(5,45)(6,46)(7,47)(8,48)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,23,51,39)(2,22,52,38)(3,21,49,37)(4,24,50,40)(5,17,36,61)(6,20,33,64)(7,19,34,63)(8,18,35,62)(9,25,53,41)(10,28,54,44)(11,27,55,43)(12,26,56,42)(13,29,57,45)(14,32,58,48)(15,31,59,47)(16,30,60,46) );
 
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),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,19),(2,18),(3,17),(4,20),(5,37),(6,40),(7,39),(8,38),(9,15),(10,14),(11,13),(12,16),(21,36),(22,35),(23,34),(24,33),(25,31),(26,30),(27,29),(28,32),(41,47),(42,46),(43,45),(44,48),(49,61),(50,64),(51,63),(52,62),(53,59),(54,58),(55,57),(56,60)], [(1,13,3,15),(2,58,4,60),(5,41,7,43),(6,26,8,28),(9,19,11,17),(10,64,12,62),(14,50,16,52),(18,54,20,56),(21,31,23,29),(22,48,24,46),(25,34,27,36),(30,38,32,40),(33,42,35,44),(37,47,39,45),(49,59,51,57),(53,63,55,61)], [(1,11),(2,12),(3,9),(4,10),(5,45),(6,46),(7,47),(8,48),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28),(29,36),(30,33),(31,34),(32,35),(37,41),(38,42),(39,43),(40,44),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,23,51,39),(2,22,52,38),(3,21,49,37),(4,24,50,40),(5,17,36,61),(6,20,33,64),(7,19,34,63),(8,18,35,62),(9,25,53,41),(10,28,54,44),(11,27,55,43),(12,26,56,42),(13,29,57,45),(14,32,58,48),(15,31,59,47),(16,30,60,46)]])
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4N4O···4AG
order122222222224···44···4
size111122224442···24···4

44 irreducible representations

dim1111111111111244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C4○D42- 1+4C2.C25
kernelC22.104C25C4×C4○D4C23.33C23C2×C42.C2C23.36C23C23.37C23C22.33C24C22.35C24D4⋊6D4C22.46C24C22.47C24D4⋊3Q8Q8⋊3Q8D4C4C2
# reps1122314238221822

Matrix representation of C22.104C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
330000
420000
000330
003003
000002
000020
,
100000
010000
000200
003000
000003
000020
,
100000
340000
002000
000200
000130
001003
,
100000
010000
000100
001000
000001
000010
,
300000
030000
003000
000300
000420
004002

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

C22.104C25 in GAP, Magma, Sage, TeX 

C_2^2._{104}C_2^5
 
% in TeX
 
G:=Group("C2^2.104C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2247);
 
// by ID
 
G=gap.SmallGroup(128,2247);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,184,570,1684,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=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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