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G = C22.88C25  order 128 = 27

69th central stem extension by C22 of C25

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

Aliases: C4⋊22- (1+4), C23.44C24, C22.88C25, C42.576C23, D4○(C4⋊Q8), C4○D4⋊11D4, (D4×Q8)⋊18C2, Q8○(C4⋊1D4), D4.59(C2×D4), Q8.61(C2×D4), D4⋊6D4⋊22C2, (C2×C4).78C24, C2.34(D4×C23), C4⋊C4.294C23, C4.123(C22×D4), C4⋊Q8.340C22, (C4×D4).231C22, (C2×D4).471C23, C22⋊C4.23C23, (C2×2- (1+4))⋊9C2, (C4×Q8).327C22, (C2×Q8).448C23, C22.15(C22×D4), C4⋊D4.225C22, C4⋊1D4.195C22, (C2×C42).943C22, (C22×C4).360C23, C22⋊Q8.114C22, C2.23(C2×2- (1+4)), C22.26C24⋊37C2, C4.4D4.175C22, (C22×Q8).359C22, C22.D4.7C22, C42⋊C2.344C22, C23.38C23⋊22C2, (C2×D4)○(C4⋊Q8), (C2×C4⋊Q8)⋊55C2, (C4×C4○D4)⋊27C2, (C2×Q8)○(C4⋊1D4), (C2×C4).188(C2×D4), (C2×C4⋊C4).703C22, (C2×C4○D4).228C22, SmallGroup(128,2231)

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

Derived series
C1 — C22 — C22.88C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.88C25
Lower central
C1 — C22 — C22.88C25
Upper central
C1 — C22 — C22.88C25
Jennings
C1 — C22 — C22.88C25

Subgroups: 1052 in 738 conjugacy classes, 432 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C4 [×18], C22, C22 [×6], C22 [×18], C2×C4, C2×C4 [×35], C2×C4 [×60], D4 [×12], D4 [×36], Q8 [×4], Q8 [×36], C23 [×7], C42, C42 [×9], C22⋊C4 [×30], C4⋊C4 [×42], C22×C4 [×33], C2×D4 [×21], C2×Q8, C2×Q8 [×18], C2×Q8 [×40], C4○D4 [×8], C4○D4 [×72], C2×C42 [×3], C2×C4⋊C4 [×12], C42⋊C2 [×3], C4×D4 [×18], C4×Q8 [×2], C4⋊D4 [×12], C22⋊Q8 [×24], C22.D4 [×24], C4.4D4 [×6], C4⋊1D4, C4⋊Q8 [×21], C22×Q8 [×10], C2×C4○D4, C2×C4○D4 [×18], 2- (1+4) [×16], C4×C4○D4, C2×C4⋊Q8 [×3], C22.26C24 [×3], C23.38C23 [×6], D4⋊6D4 [×12], D4×Q8 [×4], C2×2- (1+4) [×2], C22.88C25

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2- (1+4) [×4], C25, D4×C23, C2×2- (1+4) [×2], C22.88C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=d2=f2=1, c2=g2=a, e2=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 >

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
140000
000200
002000
000002
000020
,
100000
010000
000300
002000
000002
000030
,
130000
140000
004000
000400
000040
000004
,
400000
040000
000010
000001
001000
000100
,
100000
010000
000100
004000
000001
000040

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4L4M···4AD
order12222···222224···44···4
size11112···244442···24···4

44 irreducible representations

dim1111111124
type+++++++++-
imageC1C2C2C2C2C2C2C2D42- (1+4)
kernelC22.88C25C4×C4○D4C2×C4⋊Q8C22.26C24C23.38C23D4⋊6D4D4×Q8C2×2- (1+4)C4○D4C4
# reps11336124284

In GAP, Magma, Sage, TeX 

C_2^2._{88}C_2^5
 
% in TeX
 
G:=Group("C2^2.88C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2231);
 
// by ID
 
G=gap.SmallGroup(128,2231);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,352,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=g^2=a,e^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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