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G = C24.216C23  order 128 = 27

56th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.216C23, C23.244C24, C22.582- (1+4), C4.74(C4×D4), C4⋊C4.395D4, C22⋊Q8⋊20C4, C2.7(D4⋊6D4), C2.6(Q8⋊5D4), (C2×C42).23C22, C23.94(C22×C4), C22.135(C23×C4), (C22×C4).766C23, (C23×C4).309C22, C22.115(C22×D4), C24.C22.7C2, (C22×Q8).405C22, C23.67C23⋊21C2, C23.65C23⋊27C2, C23.63C23⋊19C2, C2.C42.63C22, C2.5(C22.50C24), C2.8(C22.46C24), C2.17(C23.32C23), (C2×C4×Q8)⋊9C2, (C4×C4⋊C4)⋊44C2, C2.38(C2×C4×D4), C2.34(C4×C4○D4), C4⋊C4.157(C2×C4), (C2×C4).890(C2×D4), C22⋊C4.11(C2×C4), (C4×C22⋊C4).30C2, (C2×C4).43(C22×C4), (C2×Q8).150(C2×C4), (C2×C22⋊Q8).19C2, (C2×C4).800(C4○D4), (C2×C4⋊C4).976C22, (C22×C4).315(C2×C4), C22.129(C2×C4○D4), (C2×C42⋊C2).34C2, (C2×C22⋊C4).443C22, SmallGroup(128,1094)

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

Derived series
C1 — C22 — C24.216C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C24.216C23
Lower central
C1 — C22 — C24.216C23
Upper central
C1 — C23 — C24.216C23
Jennings
C1 — C23 — C24.216C23

Subgroups: 444 in 288 conjugacy classes, 152 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×22], C22 [×7], C22 [×10], C2×C4 [×24], C2×C4 [×38], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×16], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×16], C4⋊C4 [×10], C22×C4 [×6], C22×C4 [×12], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C42 [×6], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C42⋊C2 [×4], C4×Q8 [×4], C22⋊Q8 [×8], C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C4×C4⋊C4 [×2], C23.63C23 [×2], C24.C22 [×4], C23.65C23, C23.67C23, C2×C42⋊C2, C2×C4×Q8, C2×C22⋊Q8, C24.216C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×6], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×3], 2- (1+4) [×2], C2×C4×D4, C4×C4○D4, C23.32C23, D4⋊6D4, Q8⋊5D4, C22.46C24, C22.50C24, C24.216C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=d, g2=b, faf-1=ab=ba, eae-1=ac=ca, ad=da, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Smallest permutation representation
►On 64 points
Generators in S64
(2 52)(4 50)(5 62)(6 39)(7 64)(8 37)(10 22)(12 24)(14 26)(16 28)(17 45)(18 58)(19 47)(20 60)(29 57)(30 46)(31 59)(32 48)(33 63)(34 40)(35 61)(36 38)(42 54)(44 56)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
001000
001400
000010
000044
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
010000
400000
001300
000400
000020
000033
,
200000
020000
001000
000100
000043
000011
,
400000
010000
001000
001400
000020
000002

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,2,3,0,0,0,0,0,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
 

50 conjugacy classes

class 1 2A···2G2H2I4A···4X4Y···4AN
order12···2224···44···4
size11···1442···24···4

50 irreducible representations

dim11111111111224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42- (1+4)
kernelC24.216C23C4×C22⋊C4C4×C4⋊C4C23.63C23C24.C22C23.65C23C23.67C23C2×C42⋊C2C2×C4×Q8C2×C22⋊Q8C22⋊Q8C4⋊C4C2×C4C22
# reps1132411111164122

In GAP, Magma, Sage, TeX 

C_2^4._{216}C_2^3
 
% in TeX
 
G:=Group("C2^4.216C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1094);
 
// by ID
 
G=gap.SmallGroup(128,1094);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=d,g^2=b,f*a*f^-1=a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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