C2^3.4D28 - GroupNames

G = C₂³.₄D₂₈ order 448 = 2⁶·7

4^th non-split extension by C₂³ of D₂₈ acting via D₂₈/C₇=D₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₄D₂₈, C₄.D₄.D₇, (C₂×D₄).₆D₁₄, (C₂×C₂₈).₁₄D₄, C₂³.D₇⋊₄C₄, C₂³.₄(C₄×D₇), (C₂²×Dic₇)⋊₂C₄, (C₂²×C₁₄).₁₃D₄, C₇⋊₂(C₂³.D₄), C₂³⋊Dic₇.₃C₂, C₁₄.₁₂(C₂³⋊C₄), C₂².₁₃(D₁₄⋊C₄), (D₄×C₁₄).₁₇₁C₂², C₂³.₁₈D₁₄.₄C₂, C₂.₁₃(C₂³.₁D₁₄), (C₂×C₄).₂(C₇⋊D₄), (C₂²×C₁₄).₄(C₂×C₄), (C₇×C₄.D₄).₁C₂, (C₂×C₁₄).₆(C₂²⋊C₄), SmallGroup(448,33)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — C₂³.₄D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — D₄×C₁₄ — C₂³.₁₈D₁₄ — C₂³.₄D₂₈

Lower central

C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂³.₄D₂₈

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₄.D₄

Generators and relations for C₂³.₄D₂₈
G = < a,b,c,d,e | a²=b²=c²=1, d²⁸=c, e²=ca=ac, ab=ba, dad^-1=abc, ae=ea, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=acd²⁷ >

Subgroups: 396 in 68 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₄.D₄, C₂².D₄, C₅₆, C₂×Dic₇, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₂³.D₄, Dic₇⋊C₄, C₂³.D₇, C₂³.D₇, C₇×M₄(2), C₂²×Dic₇, D₄×C₁₄, C₂³⋊Dic₇, C₇×C₄.D₄, C₂³.₁₈D₁₄, C₂³.₄D₂₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₇, C₂²⋊C₄, D₁₄, C₂³⋊C₄, C₄×D₇, D₂₈, C₇⋊D₄, C₂³.D₄, D₁₄⋊C₄, C₂³.₁D₁₄, C₂³.₄D₂₈

Smallest permutation representation of C₂³.₄D₂₈
►On 112 points

Generators in S₁₁₂

(1 67)(2 96)(3 97)(4 70)(5 71)(6 100)(7 101)(8 74)(9 75)(10 104)(11 105)(12 78)(13 79)(14 108)(15 109)(16 82)(17 83)(18 112)(19 57)(20 86)(21 87)(22 60)(23 61)(24 90)(25 91)(26 64)(27 65)(28 94)(29 95)(30 68)(31 69)(32 98)(33 99)(34 72)(35 73)(36 102)(37 103)(38 76)(39 77)(40 106)(41 107)(42 80)(43 81)(44 110)(45 111)(46 84)(47 85)(48 58)(49 59)(50 88)(51 89)(52 62)(53 63)(54 92)(55 93)(56 66)

(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)

(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(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)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)

(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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)

(1 108 95 42)(2 41 68 79)(3 106 69 12)(4 11 98 77)(5 104 99 38)(6 37 72 75)(7 102 73 8)(9 100 103 34)(10 33 76 71)(13 96 107 30)(14 29 80 67)(15 94 81 56)(16 55 110 65)(17 92 111 26)(18 25 84 63)(19 90 85 52)(20 51 58 61)(21 88 59 22)(23 86 89 48)(24 47 62 57)(27 82 93 44)(28 43 66 109)(31 78 97 40)(32 39 70 105)(35 74 101 36)(45 64 83 54)(46 53 112 91)(49 60 87 50)

G:=sub<Sym(112)| (1,67)(2,96)(3,97)(4,70)(5,71)(6,100)(7,101)(8,74)(9,75)(10,104)(11,105)(12,78)(13,79)(14,108)(15,109)(16,82)(17,83)(18,112)(19,57)(20,86)(21,87)(22,60)(23,61)(24,90)(25,91)(26,64)(27,65)(28,94)(29,95)(30,68)(31,69)(32,98)(33,99)(34,72)(35,73)(36,102)(37,103)(38,76)(39,77)(40,106)(41,107)(42,80)(43,81)(44,110)(45,111)(46,84)(47,85)(48,58)(49,59)(50,88)(51,89)(52,62)(53,63)(54,92)(55,93)(56,66), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(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)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,108,95,42)(2,41,68,79)(3,106,69,12)(4,11,98,77)(5,104,99,38)(6,37,72,75)(7,102,73,8)(9,100,103,34)(10,33,76,71)(13,96,107,30)(14,29,80,67)(15,94,81,56)(16,55,110,65)(17,92,111,26)(18,25,84,63)(19,90,85,52)(20,51,58,61)(21,88,59,22)(23,86,89,48)(24,47,62,57)(27,82,93,44)(28,43,66,109)(31,78,97,40)(32,39,70,105)(35,74,101,36)(45,64,83,54)(46,53,112,91)(49,60,87,50)>;

G:=Group( (1,67)(2,96)(3,97)(4,70)(5,71)(6,100)(7,101)(8,74)(9,75)(10,104)(11,105)(12,78)(13,79)(14,108)(15,109)(16,82)(17,83)(18,112)(19,57)(20,86)(21,87)(22,60)(23,61)(24,90)(25,91)(26,64)(27,65)(28,94)(29,95)(30,68)(31,69)(32,98)(33,99)(34,72)(35,73)(36,102)(37,103)(38,76)(39,77)(40,106)(41,107)(42,80)(43,81)(44,110)(45,111)(46,84)(47,85)(48,58)(49,59)(50,88)(51,89)(52,62)(53,63)(54,92)(55,93)(56,66), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(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)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,108,95,42)(2,41,68,79)(3,106,69,12)(4,11,98,77)(5,104,99,38)(6,37,72,75)(7,102,73,8)(9,100,103,34)(10,33,76,71)(13,96,107,30)(14,29,80,67)(15,94,81,56)(16,55,110,65)(17,92,111,26)(18,25,84,63)(19,90,85,52)(20,51,58,61)(21,88,59,22)(23,86,89,48)(24,47,62,57)(27,82,93,44)(28,43,66,109)(31,78,97,40)(32,39,70,105)(35,74,101,36)(45,64,83,54)(46,53,112,91)(49,60,87,50) );

G=PermutationGroup([[(1,67),(2,96),(3,97),(4,70),(5,71),(6,100),(7,101),(8,74),(9,75),(10,104),(11,105),(12,78),(13,79),(14,108),(15,109),(16,82),(17,83),(18,112),(19,57),(20,86),(21,87),(22,60),(23,61),(24,90),(25,91),(26,64),(27,65),(28,94),(29,95),(30,68),(31,69),(32,98),(33,99),(34,72),(35,73),(36,102),(37,103),(38,76),(39,77),(40,106),(41,107),(42,80),(43,81),(44,110),(45,111),(46,84),(47,85),(48,58),(49,59),(50,88),(51,89),(52,62),(53,63),(54,92),(55,93),(56,66)], [(2,30),(4,32),(6,34),(8,36),(10,38),(12,40),(14,42),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(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),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,108,95,42),(2,41,68,79),(3,106,69,12),(4,11,98,77),(5,104,99,38),(6,37,72,75),(7,102,73,8),(9,100,103,34),(10,33,76,71),(13,96,107,30),(14,29,80,67),(15,94,81,56),(16,55,110,65),(17,92,111,26),(18,25,84,63),(19,90,85,52),(20,51,58,61),(21,88,59,22),(23,86,89,48),(24,47,62,57),(27,82,93,44),(28,43,66,109),(31,78,97,40),(32,39,70,105),(35,74,101,36),(45,64,83,54),(46,53,112,91),(49,60,87,50)]])

46 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	7A	7B	7C	8A	8B	14A	14B	14C	14D	14E	14F	14G	···	14L	28A	···	28F	56A	···	56L
order	1	2	2	2	2	4	4	4	4	4	4	7	7	7	8	8	14	14	14	14	14	14	14	···	14	28	···	28	56	···	56
size	1	1	2	4	4	4	28	28	56	56	56	2	2	2	8	8	2	2	2	4	4	4	8	···	8	4	···	4	8	···	8

46 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+	+			+	+			-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₇	D₁₄	C₇⋊D₄	C₄×D₇	D₂₈	C₂³⋊C₄	C₂³.D₄	C₂³.₁D₁₄	C₂³.₄D₂₈
kernel	C₂³.₄D₂₈	C₂³⋊Dic₇	C₇×C₄.D₄	C₂³.₁₈D₁₄	C₂³.D₇	C₂²×Dic₇	C₂×C₂₈	C₂²×C₁₄	C₄.D₄	C₂×D₄	C₂×C₄	C₂³	C₂³	C₁₄	C₇	C₂	C₁
# reps	1	1	1	1	2	2	1	1	3	3	6	6	6	1	2	6	3

Matrix representation of C₂³.₄D₂₈ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	15	8	0	0
0	0	85	98	0	0
0	0	0	98	0	15
0	0	85	98	98	0

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	81	0	112	0
0	0	81	0	0	112

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

80	22	0	0	0	0
69	91	0	0	0	0
0	0	1	0	106	0
0	0	0	0	112	1
0	0	83	0	112	0
0	0	111	15	112	0

106	70	0	0	0	0
80	7	0	0	0	0
0	0	15	0	8	0
0	0	0	0	98	15
0	0	83	112	98	0
0	0	2	0	98	0

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,85,0,85,0,0,8,98,98,98,0,0,0,0,0,98,0,0,0,0,15,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,81,81,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[80,69,0,0,0,0,22,91,0,0,0,0,0,0,1,0,83,111,0,0,0,0,0,15,0,0,106,112,112,112,0,0,0,1,0,0],[106,80,0,0,0,0,70,7,0,0,0,0,0,0,15,0,83,2,0,0,0,0,112,0,0,0,8,98,98,98,0,0,0,15,0,0] >;

C₂³.₄D₂₈ in GAP, Magma, Sage, TeX

C_2^3._4D_{28}

% in TeX

G:=Group("C2^3.4D28");

// GroupNames label

G:=SmallGroup(448,33);

// by ID

G=gap.SmallGroup(448,33);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,184,346,297,851,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*a=a*c,a*b=b*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^27>;

// generators/relations

G = C23.4D28 order 448 = 26·7

4th non-split extension by C23 of D28 acting via D28/C7=D4

G = C₂³.₄D₂₈ order 448 = 2⁶·7

4^th non-split extension by C₂³ of D₂₈ acting via D₂₈/C₇=D₄